During the First Age of Middle-earth Manwë sent Thorondor and the Eagles to watch over the inhabitants including the Noldor. Throughout the First Age, Thorondor intervened on behalf of Men and Elves and even left wounds on Morgoth and was a great asset to the War of Wrath. We know that the other Eagles stayed behind during the later Ages.
What happened to Thorondor? Did he fly back to Valinor or did he perish in the War of Wrath?
The short answer is "We don't know." The longer answer is "We don't know, but here's some information which doesn't actually answer the question, but seems somewhat relevant."
It seems likely that the great Eagles (including Thorondor) were created in a scene from The Silmarillion:
Then Manwë awoke, and he went down to Yavanna upon Ezellohar, and he sat beside her beneath the Two Trees. And Manwë said 'O Kementári, Eru hath spoken, saying "Do then any of the Valar suppose that I did not hear all the Song, even the least sound of the least voice? Behold! When the Children awake, then the thought of Yavanna will awake also, and it will summon spirits from afar, and they will go among the kelvar and the olvar, and some will dwell therein, and be held in reverence, and their just anger shall be feared. For a time while the Firstborn are in their power, and while the Secondborn are young."
But dost them not now remember, Kementári, that thy thought sang not always alone? Did not thy thought and mine meet also, so that we took wing together like great birds that soar above the clouds? That also shall come to be by the heed of Ilúvatar, and before the Children awake there shall go forth with wings like the wind the Eagles of the Lords of the West.'
This suggests that the Eagles (and the Ents) were destined to remain in Middle-earth only while the Secondborn (Men) were young. When was that? Hard to say! The key point is that it almost certainly ended at the end of the Third Age when Middle-earth was turned over to Men and the remaining First-born rapidly left Middle-earth.
Gandalf said: '...The Third Age of the world is ended, and the new age is begun... For though much has been saved, much must now pass away; and the power of the Three Rings also is ended. ... For the time comes of the Dominion of Men, and the Elder Kindred shall fade or depart.'
There is no suggestion anywhere that Thorondor was immortal any more than the Ents were immortal. Both could be killed and could die. But neither died of old age or any natural cause.
The last we hear of Thorondor is at the battle against Morgoth which ended the First Age. We do hear of Eagles which would appear to be of the same sort in both The Hobbit and LotR, but not of Thorondor.
In LotR we heard of some of his descendents:
There came Gwaihir the Windlord, and Landroval his brother, greatest of all the Eagles of the North, mightiest of the descendants of old Thorondor, who built his eyries in the inaccessible peaks of the Encircling Mountains when Middle-earth was young.
So, did he die in the battle that ended the Third Age? Maybe, though it's not mentioned. Did he die during the Second or Third Ages, perhaps fighting a dragon or one of the Nazgûl or Sauron himself? Perhaps. Did he leave Middle-earth and returning to being a spirit? Could be. Is he still around during LotR, but retired from hero-saving? Maybe so.
We just don't know.
As Tolkien wrote it, this passage from The Silmarillion mentions Landroval and Gwaihir.
Thus the quest of the Silmaril was like to have ended in ruin and despair; but in that hour above the wall of the valley three mighty birds appeared, flying northward with wings swifter than the wind. Thorondor was their leader; and with him were his mightiest vassals, wide-winged Lhandroval, and Gwaihir the lord of storm. Among all birds and beasts the wandering and need of Beren had been noised, and Huan himself had bidden all things watch, that they might bring him aid. High above the realm of Morgoth Thorondor and his vassals soared, and seeing now the madness of the Wolf and Beren's fall they came swiftly down, even as the powers of Angband were released from the toils of sleep.
The Silmarillion - Chapter 19 - "Of Beren and Luthien" (reconstructed)
Landroval and Gwaihir also appear in The Lord of the Rings.
There came Gwaihir the Windlord, and Landroval his brother, greatest of all the Eagles of the North, mightiest of the descendants of old Thorondor, who built his eyries in the inaccessible peaks of the Encircling Mountains when Middle-earth was young.
The Lord of the Rings - Book VI Chapter 4 - "The Field of Cormallen"
When editing the Silmarillion, Christopher removed the bolded line, as he thought their appearance in The Lord of the Rings meant they were supposed to be removed from the Silmarillion.
Later, when he better understood the dating of the manuscripts, he realized that not only did his father continue working on the manuscript without removing them, but that Tolkien updated the spelling of the names to match the way they appeared in The Lord of the Rings, thus indicating that Tolkien felt these were the same characters.
with wings swifter than the wind (p. 182) The draft text B (see p. 293) has at this point: 'Thorondor led them, and the others were Lhandroval (Wide-wing) and Gwaewar his vassal.' In the following text C, also of 1937, this became: 'Thorondor was their leader; and with him were his mightiest vassals, wide-winged Lhandroval, and Gwaewar lord of the wind.' This was emended (in 1951, seep. 294) to 'Gwaihir the lord of storm', and in this form the passage is found in the QS manuscript. It was omitted in The Silmarillion on account of the passage in The Return of the King (VI. 4): 'There came Gwaihir the Windlord, and Landroval his brother . . . mightiest of the descendants of old Thorondor, who built his eyries in the inaccessible peaks of the Encircling Mountains when Middle-earth was young.' At the time, I did not understand the nature and dating of the end of QS. It now appears that there was no reason to suppress the names; in fact, it seems that Gwaewar was changed to Gwaihir to bring it into accord with The Lord of the Rings - however this is to be interpreted.
The Lost Road and Other Writings - Commentary to Quenta Silmarillion Chs. 12-15
If Landroval and Gwaihir are still alive, Thorondor could be as well. That said, he doesn't actually appear, so we do not know if he is or not.
