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 Manwe awoke, and he went down to Yavanna upon Ezellohar, and he sat beside her beneath the Two Trees. And Manwe said 'O Kementri, 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, Kementri, 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 Iluvatar, 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 Nazgul 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.