As far as I know, only one mountain in all of Middle-Earth or even all Arda, had a specific elevation from base to top mentioned.
The Return of the King, Book VI, Chapter 3, "Mount Doom", when Sam has carried Frodo part way up Mount Doom:
The Mountain standing ominous and alone had looked taller than it was. Sam saw now that it was less lofty than the high passes of the Ephel Duath which he and Frodo had scaled. The confused and tumbled shoulders of its great base rose for maybe three thousand feet above the plain, and above them was reared half as high again its tall central cone, like a vast ost or chimney capped with a jagged crater. But already Sam was more than halfway up the base, and the plain of Gorgoroth was dim below him, wrapped in fume and shadow.
Of course the relative and absolute dimensions of Mount Doom are not exact, since the base rose for "maybe three thosuand feet" and not for an exact figure, and the central cone was probably not precisely "half as high again" as the base. So obviously the actual height of Mount Doom at that moment could easily be several percentage points higher or lower than calculated from that statement.
But the big problem is with the cone being half as high again as the base. That can be interpreted two ways.
One) the height of the cone is about half that of the 3,000-foot-high base, again", and thus about 1,500 feet, thus making the total height of the mountain "half as high again" as the base, or 4,500 feet total.
Two) the height of the cone itself is "half as high again" as the 3,000-foot-high base, or 4,500 feet, which added to the 3,000-foot-high base, makes the total height of Mount Doom 7,5000 feet.
So at that time the crater at the top of Mount Doom should be either about 4,500 feet or about 7,500 feet above the plain.
So the passes of the Ephel Duath mountains should be higher by an unstated amount than 4,500 feet, or 7,500 feet, and the peaks should be much higher than the passes.
In The Silmarillion, the Quenta Silmarillion, Chapter III, "Of the Coming of the Elves and the Captivity of Melkor" Orome leads the Elves westward through Middle-earth on the Great Journey:
And it came to pass after many years of journeying in this manner that the Eldar took their course thorugh a forest, and they came to a great river, wider than any they had yet seen, and beyond them were mountains whose sharp horns seemed to pierce the realm of the stars. This river, it is said, was even the river which was afterwards called Anduin the Great, and was ever the frontier of the west-lands of Middle-Earth. But the mountains were the Hithaeglir, the Towers of Mist upon the borders of Eriador; yet they were taller and more terrible in those days, and were reared by Melkor to hinder the riding of Orome.
I thought that quotation said that the Misty Mountains were the highest mountain range in Middle-earth, but it doesn't. But it does say that the absolute heights of the Misty Mountains and no doubt other mountains, and thus their relative height rankings, changed in various convulsions of the earth between the Eldar days and the composing of the Quenta Silmarillion.
So the tallest mountains in Middle-Earth did change from age to age.