As I wrote in the comments of @user8078's answer, Rabbi Reuven Margaliyot wrote an essay called "חכמי התלמוד ותרגום השבעים" (The Talmudic Sages and The LXX). In it he brings eight examples of inexplicit references he found to the LXX in the Talmud and Midrashim. All are cases in which the sages asserted a certain understanding, spelling or vowelization over another, the other option being something from the LXX. Among the examples are a number from the Prophets and Writings.
Based on his work, I thought I would examine the Chazalic texts on the sections of the Masoretic Jeremiah that are missing from the LXX. I will note a few things to bear in mind with my research:
a. I searched for a list of the differences between the MT Jeremiah and the LXX Jeremiah. I found a few sites that bring lists but weren't exactly the same. I used the list I found on ellops together with another section mentioned in bible.ca.
b. While it's traditionally and academically thought that the Septuagint was written/translated a couple of centuries before Jesus and the rise of Christianity, it is known that later Christians deliberately subjectively edited it. According to Jerome, these at least included Hesychus, Origen and Lucian. Rabbi Dr. M. Z. Segal also pointed out in מבוא המקרא (Introduction to the Bible), vol. II that it is very likely that later individuals edited the LXX's Jeremiah, further differentiating it from the theoretical original Hebrew version. From a Jewish perspective, this would make the LXX corrupted and no longer valid. Bible.ca in the above linked article assert that the original Hebrew version used by the first authors of the LXX Jeremiah was also in the hands of one of the Dead Sea cults, evidence that the LXX's version was in circulation at least around the time the MT was in circulation. However, they say this based on a single Jeremiah DSS fragment that is is missing one of the sections that are also missing in the LXX version. This validates that part of the LXX, but does not validate the other parts. Furthermore, the rest of the Jeremiah DSS fragments, from what I've gathered, are MT-like (I've also heard that bible.ca is an untrustworthy site, which seems to line up with their claim that only four sections from the MT Jeremiah are missing in the LXX).
My research was therefore based on the assumption that the sages were aware of a corrupted LXX Jeremiah, corrupted by the early Christians. Evidence to the contrary would probably invalidate most or all of the following.
c. It's much more complicated to compare a text that says X with another that is simply missing X than what Rabbi Margaliyot did, which is to compare a text that says X with a text that says Y, so the result is that this is mostly speculation on my part. Out of my various speculations, I narrowed the list down to the ones that I think are most likely to have been hints from the sages towards the existence of a corrupted translation of Jeremiah in circulation throughout the Roman Empire.
"All nations shall serve him, his son and his grandson—until the turn of his own land comes, when many nations and great kings shall subjugate him."
Esther Rabbah 3:5: "Another explanation: "You do look! You take note of mischief and vexation!": the congregation of Israel said before the Holy One, blessed be He: "Master of the universe, "You do look!" at Nebuchadnezzar the wicked man who came and destroyed the Temple and exiled Israel and chained them in collars. "To requite is in Your power" and you rested your shekhinah on Jeremiah and he said to us: "All nations shall serve him, his son and his grandson (Jeremiah 27:7)". "To You the hapless can entrust himself" tomorrow he comes and takes Hananiah, Mishael, and Azariah and will place them in the midst of a fiery furnace and will say to them: "What god is there that can save you from my power? (Daniel 3:15)". Rather "You have ever been the orphan’s help": a certain orphan was left to him and you made her the augusta empress over a kingdom that was not hers. Who? She was Vashti."
According to the midrash, the verse comes to tell us that all of the hardships that befell Israel came directly from God; God watches over the world and all those who rise to power will eventually fall - for example, the orphan girl who had nothing was adopted by Nevuchadnezzar and eventually came to rule over Babylon, upon becoming Queen of Persia - Vashti. This is can be seen as a hint from the sages that even Rome will fall eventually, so people (such as the Christians who at the time attempted to convert the Romans) should not put their faith in Rome but in God, like Mishael, Chananyah and Azaryah did.
"But you, Have no fear, My servant Jacob —declares the LORD— Be not dismayed, O Israel! I will deliver you from far away, Your folk from their land of captivity. And Jacob shall again have calm And quiet with none to trouble him"
Tractate Taanit 5b: "After they had eaten, Rabbi Yitzḥak said to Rav Naḥman that Rabbi Yoḥanan said as follows: Our patriarch Jacob did not die. Rav Naḥman asked him in surprise: And was it for naught that the eulogizers eulogized him and the embalmers embalmed him and the buriers buried him? Rabbi Yitzḥak replied to Rav Naḥman: I am interpreting a verse, as it is stated: “Therefore do not fear, Jacob My servant, says the Lord, neither be dismayed, Israel, for I will save you from afar, and your seed from the land of their captivity” (Jeremiah 30:10). This verse juxtaposes Jacob to his seed: Just as his seed is alive when redeemed, so too, Jacob himself is alive."
The Talmud here says that the verse implies that the People of Israel are still alive, by contrast to the Christian belief that Israel is a dead nation, replaced by a "spiritual Israel".
Tanchuma Buber, Vayikra 17:1: "...All the generations, the generation of Enosh, the generation of the flood, and the generation of the dispersion (i.e., of the Tower of Babel), did not learn from each other. Instead they were exalting themselves. Therefore, (in Is. 57:20): BUT THE WICKED ARE LIKE THE TROUBLED SEA. The wicked have no rest in the world, but the righteous have serenity (ShQT), as stated (in Jer. 30:10): AND JACOB SHALL AGAIN HAVE PEACE (ShQT) AND QUIET WITH NONE TO MAKE HIM AFRAID."
The wicked are compared to a troubled sea: like a wave that rises, believing it is the greatest, not knowing that it is about to break against the beach, because it was too prideful to ask the previous wave, so too are the wicked who didn't ask for advice from the previous generations.
This last one seems like a pretty clear-cut jab at either Christians in general or the LXX authors specifically, because like the wicked people in the parable, they did not seek the advice of previous generations (the Jewish traditions which came before Christianity and the MT text) but did whatever they pleased, not knowing that they too, will eventually fall.
Pesikata D'Rav Kahana 23:2: "He dreamed, and behold, a ladder set upon the earth; and the top of it reached to heaven; and behold, the angels of God ascending and descending on it (Gen. 28:12). R. Samuel the son of Nahman declared: These were the guardian angels of the idolatrous nations. He explained further: This verse teaches us that the Holy One, blessed be He, showed Jacob, our father, the guardian angel of Babylon ascending seventy rungs of the ladder and descending, the guardian angel of Media ascending fifty-two rungs of the ladder and descending, the guardian angel of Greece ascending one hundred rungs of the ladder and descending, and the guardian angel of Edom ascending the ladder. Jacob did not know how many rungs of the ladder the guardian angel of Edom mounted, and he therefore cried out in terror: Perhaps Edom will never be compelled to descend. The Holy One, blessed be He, then said to him: Fear thou not, O Jacob, My servant; neither be dismayed, O Israel (Jer. 30:10), for even though you see him ascending unto My throne as though that were possible, I will cast him down, as is said: Though thou wert to rise as high as the eagle, and though thou set thy nest among the stars, I will bring thee down from thence, saith the Lord (Obad. 4)...said to him the Holy [One], Jacob, you too shall climb up, at that hour our father Jacob was fearful, [and] said: Perhaps like these had a descent, I too shall descend? Said to him the Holy One, Blessed is He, "Be not dismayed, O Israel!", if you climb, you will not descend forever, and he didn't believe and didn't climb..."
The first part, about the different nations climbing the ladder, is a well-known midrash that appears in multiple sources. Obviously, it foretells the fall of Rome. Like in Esther Rabbah, the idea is that one shouldn't put their faith in Rome but in God. The second part, of Jacob's fear appears to be a jab at people who do not put their faith in God and doubt Israel's ability to one day rise once again and instead turn to other places for salvation (the Christians notably started to focus most of their efforts on converting non-Jews).
"In those days and at that time, I will raise up a true branch of David’s line, and he shall do what is just and right in the land."
Targum Jonathan: "In that day and that time I will raise up of David a righteous anointed one [messiah], and he shall do what is just and right in the land."
I don't know when exactly Yonatan ben Uziel, author of the Targum, lived, nor when he exactly wrote the Targum, but as he was part of the first generation of Tannaim, he may have lived around Jesus's time. If so, the assertion in his translation that the name of the messiah has yet to be revealed (as he hasn't arrived yet) would be a denial of Jesus's purported messianic status.
"Thus said the LORD: As surely as I have established My covenant with day and night—the laws of heaven and earth."
a. Avodah Zara 3a, Nedarim 32a, Pesachim 68b, Shabbat 33a, Bamidbar Rabbah 10:1, Midrash Lekach Tov Beresheet 1:1:29, Tanchuma Haazinu 3:1, Midrash Tehillim 20:1, Esther Rabbah 7:13, Otzar Hamidrashim Chupat Eliyahu, Chupat Eliyahu Rabbah 265, Pirkei D'Rabbi Eliezer 16:2 - all of these sources assert that the covenant in this verse refers to the Torah; the study of Torah. The various sources come to explain that the Torah is a very basic and essential thing in the world, something that the world couldn't exist without. This is in contrast to the Christian perspective which removed much of the importance of the Torah, based on their understanding of Jeremiah's "new covenant".
b. Mishnah Nedarim 3:11, Tosefta Nedarim 2:6, Tanchuma Lech Lecha 20:4, Midrash Tehillim 6:1, Otzar Hamidrashim, Last Kundris from Midrash Yelamdenu 8, Shemot Rabbah 47:4, Tanah Devei Eliyahu Zutta, Additions to Seder Eliyahu Zutta, Pirkei D'Rabbi Eliezer 5 - all of these sources assert that the covenant in this verse refers to the covenant through the circumcision. This is in contrast to the Christian perspective which removed the need for a physical circumcision, based on their understanding of Jeremiah's "circumcision of the heart".
c. Midrash Tehillim 109:72, Otzar Hamidrashim, Midrashim on Megillat Esther, Drash for Purim 8 - these sources don't explain which covenant the verse is talking about, but they assert that this covenant was made with the People of Israel and the People of Israel are key to the existence of the whole world.
"But I will restore the fortunes of Moab in the days to come—declares the LORD. Thus far is the judgment on Moab."
Otzar Hamidrashim, Aggadat Mashiach 4: "It smashes the brow of Moab. Said R' Huna in the name of R' Levi, this teaches us that Israel were gathered in the Upper Galilee and watching over them will be Messiah ben Yosef from inside the Galilee, and they rise up from there and all of Israel with him to Jerusalem...and he ascends and builds the Temple and sacrifices sacrifices and the fire descends from heaven, and he crushes the nations of the world...and at the end makes peace with Moab, as it says "But I will restore the fortunes of Moab", and they sit for forty years in peace eating and drinking...and after all of this hear Gog and Magog and rises against them...and he enters and kills him [the messiah] in the streets of Jerusalem...and God rises and fights them...and Israel come to Rome...and march around the city and blow the horns, and in the seventh time they cry Hear O Israel the Lord our God, the Lord is One, and the city wall falls...and Israel ask for their God and their king, David, and immediately appears before them their King Messiah..." (Note: This is a long midrash so I only translated the relevant sections)
While this is a Tannaic midrash, in styling it seems like a later midrash, so it might be a later one, but it's interesting as a contrast to the Christian attempt to make Jesus out to be both Messiah ben Yosef and Messiah ben David. The midrash asserts that Messiah ben Yosef will rise from the Galilee - as Jesus did - but will gather all of Israel, build the Temple and defeat most of Israel's enemies. Jesus, of course, didn't do any of this. All he did was come from the Galilee to Jerusalem, with a small group of people from Israel. And Jesus of course died, so he can't be the Messiah ben David who will come much later.
"Concerning Edom. Thus said the LORD of Hosts: Is there no more wisdom in Teman? Has counsel vanished from the prudent? Has their wisdom gone stale?"
Chagigah 5b: "The Gemara relates: When Rabbi Yehoshua ben Ḥananya was dying, the Sages said to him: What will become of us, from the threat of the heretics, when there is no scholar like you who can refute them? He said to them that the verse states: “Is wisdom no more in Teiman? Has counsel perished from the prudent? Has their wisdom vanished?” (Jeremiah 49:7). He explained: Since counsel has perished from the prudent, from the Jewish people, the wisdom of the nations of the world has vanished as well, and there will be no superior scholars among them."
The assertion here is that there is no wisdom among the nations. This seems to be a jab at whoever searches for wisdom among the nations, such as Hellenistic Jews and Christians.
"But it is I who have bared Esau, Have exposed his place of concealment; He cannot hide. His offspring is ravaged, His kin and his neighbors— He is no more."
Beresheet Rabbah 82:12: "And Oholibamah bore etc, and over there it is written: "How thoroughly rifled is Esau", Rabbi Simon said: an onion peel, such as this for what purpose? "Have exposed his place of concealment", in order to reveal the bastards among them, and how many bastards did he bring about? Rav said three, Rabbi Levi said four, said Rabbi Binyamin in the name of Rabbi Levi, Korach over here was a bastard."
Edom is considered the spiritual ancestor of Rome (and some say also the physical ancestor; with one of the sons of Esau founding Rome). This midrash is essentially saying that Edom/Rome are a degenerate people (the idea being that the status of being a bastard is not a materialistic/physical deficiency that can be seen by all, but is actually a spiritual one hidden within the person. It can only be revealed if someone knowledgeable tells the world about how that person was born).
"This is the number of those whom Nebuchadrezzar exiled in the seventh year: 3,023 Judeans."
Sifrei Devarim 43:32: "Variantly: "and you will go lost quickly": exile after exile. And thus do you find with the ten tribes — exile after exile. And thus do you find with the tribes of Judah and Benjamin — exile after exile. They were exiled in the seventh year of Nevuchadnezzar, in the eighteenth year, and in the twenty-third year (viz. Jeremiah 52:28). R. Yehoshua b. Karcha says: An analogy: Robbers enter one's field and cut down his standing grain — and he does not protest. They cut down his sheaves — and he does not protest — until they fill up their tubs and leave. And thus is it written (Isaiah 8:23) "For there is no fatigue in (the nation, Ashur) which afflicts it (Eretz Yisrael). (This exile, the middle one,) will be (relatively) light, as (that of) the land of Zevulun and the land of Naftali (For in the second exile, too, only the two and a half tribes in trans-Jordan were exiled), but (in) the last, he will sweep (everything away).""
It's possible that the analogy brought here is supposed to be an inexplicit criticism of those people who do not preserve all of the details - like the authors of the LXX, who erased details such as this one from the LXX - much like the owner of the house who didn't pay attention. When one doesn't pay attention or protest while things are removed/stolen, in the end, nothing will be left.