The title pope (papa) was, as has been stated, at one time employed with far more latitude. In the East it has always been used to designate simple priests. In the Western Church, however, it seems from the beginning to have been restricted to bishops (Tertullian, On Modesty 13). It was apparently in the fourth century that it began to become a distinctive title of the Roman Pontiff. Pope Siricius (d. 398) seems so to use it (Ep. vi in P.L., XIII, 1164), and Ennodius of Pavia (d. 473) employs it still more clearly in this sense in a letter to Pope Symmachus (P.L., LXIII, 69). Yet as late as the seventh century St. Gall (d. 640) addresses Desiderius of Cahors as papa (P.L., LXXXVII, 265). Gregory VII finally prescribed that it should be confined to the successors of Peter.
Make the Borgia Papacy even worse that it was in OTL. The Pope remains as a figure head, but the job of actually ruling the Church falls to the College of Cardinals.
Exactly what it says in the title. What would it take for the Papacy to come to an end, assuming a POD after Constantine? This doesn't have to be an end to the Vatican, or the Catholic Church, just the title of Pope.
Extra points for recency.
Nuclear war happens between the USA and the USSR, and the Vatican is annihilated (by the USSR), along with most of the world's population. The survivors are then too focused on surviving to appoint a pope.
Nuclear war is officially cheating at AHCs.
Maybe somewhere between the 15th and 19th Centuries the College of Cardinals takes over completely, and the Church becomes constitutional.
Better, that he never came anywhere near Rome, and that I Peter 5:13 refers to literal Babylon...Conclusive proof emerges that St. Peter never left the Holy Land.
The simple way (in theory): Have the Islamic conquerors invade Rome and the rest of Italy. Then have some overzealous caliph to stamp out Christianity in Rome, rendering the religion extinct in the city. Et voilà! If there's no Bishop of Rome, there's no Pope.
Make the Papal office so defamated that nobody on their right mind would claim it (hard to do, there were VERY bad popes and nobody dared to abolish the papacy).
An easier way: spread the Reformation to Italy and splinter the Church in more denominations than OTL. Or make an earlier Reformation in the Middle Ages or so, make a popular Holy Roman Emperor (he would practically have to be a saint) put his weight behind it, and overthrow a VERY corrupt Pope.