The Roman Empire had many different avatars or incarnations, and thus it fell on many different dates.
I do not count the Roman Catholic Church, the Eastern Orthodox Church, or the Ottoman Empire, the Russian Empire, the French Empire, the German Empire, or the Austrian Empire as any sort of continuation of the Roman Empire.
If the leaders of those organizations made a serious claim to be heirs of the Roman Empire, they would have changed the names of their organizations to Roman Empire, and they would have changed their title to Emperor of the Romans. Since they didn't even bother to take those small steps, they didn't take any claims they might have made to be heirs of the Roman Empire very seriously, and so there is no reason for us to take those claims seriously.
My post number 159 on page 16 of that thread lists 33 possible dates for the end of the Roman Empire:
My latest list of possible dates for the fall of various parts, fragments, branches, or avatars of the Roman Empire.
273 AD: The breakaway "Palmyrene Empire" is reconquered by the genuine original Roman Empire under Aurelian.
274 AD: The breakaway "Gallic Empire" is reconquered by the genuine original Roman Empire under Aurelian.
296 AD: The breakaway empire in Britain is reconquered by the genuine original Roman Empire under Constantius.
476 AD: Western Roman Emperor Romulus Augustulus is deposed by Odoacar.
480 AD: Western Roman Emperor Julius Nepos is assassinated in Dalmatia, possibly by men paid by Odoacar and/or former Emperor Glycerius.
486 AD: "Kingdom of Soissons", Roman rump state in northern Gaul, is conquered by the Franks under Clovis.
887 AD: Charles III "The Fat" is deposed as Carolingian emperor. There is no successor emperor until 891.
899 AD: Carolingian Emperor Arnulph dies. There is no successor emperor until 901.
905 AD: Carolingian Emperor Louis III "The Blind" is deposed. There is no successor emperor until 915.
924 AD: Carolingian Emperor Berenger I is assassinated. There is no successor emperor until 962.
1018 AD: The "First Bulgarian Empire" whose rulers sometimes used the title of "Emperor of the Bulgarians and the Romans" is finally completely reconquered by the genuine original Roman Empire Under Basil II.
1204 AD: Constantinople, capital of the genuine original Roman Empire is captured by the Fourth Crusade. Several rival Roman realms develop.
1242 AD: John Komnenos Doukas, emperor at Thessalonika, submits to the overlordship of rival emperor John Doukas Vatatzes at Nicaea and uses the title of Despot instead.
1261 AD: Forces of the "Empire of Nicaea" capture Constantinople, capital of the Empire of Romania or Latin Empire of Constantinople. Most sources say that the Latin Empire ends in 1261 but many fiefdoms in it remain vassals of the Latin Emperors. This could also be considered the end of the separate "Nicene Empire" since it now restores the eastern Roman or "Byzantine" Empire.
1282/83 AD: The English conquest of Gwynedd, last independent Welsh state, whose rulers were the rightful Kings of the Britons, and who may also have been the the rightful northwestern Roman Emperors, heirs of Constantine III.
1308 AD: Mesud III, last Seljuk Sultan of Rum (Rome) is assassinated.
1371 AD: Stefan Uros V, "Emperor of the Serbians and the Romans" dies without a successor.
C. 1373 AD: John Uros, reigning in Thessaly as rival "Emperor of the Serbians and the Romans" abdicates and becomes a monk, dying in 1422 or 1423. His successor in Thessaly, Alexious Angelos Philanthropenos, does not use the imperial title.
1383 AD: James of Baux, titular Emperor of Romania, dies childless. He wills his title to Duke Louis I of Anjou who never uses the title. The closest genealogical heir to the Latin Empire is John of Artois (1321-1387) who never claims the title either. Several fiefs of the Latin Empire remain for generations without any titular emperor.
1395 AD: Ivan Shisman, "Emperor of the Bulgarians and the Romans", is deposed by the Ottomans and Bulgaria annexed. Almost total end of the Second Bulgarian Empire.
1396 AD: Ivan Stratsimir, rival "Emperor of the Bulgarians and the Romans" at Vidin, is deposed and his lands annexed by the Ottomans.
1422 AD: Constantine II, son of Ivan Stratsimir and titular "Emperor of the Bulgarians and the Romans" who sometimes ruled pieces of Bulgaria, dies in exile at the Hungarian court.
1444 AD: Crusade of Varna. Fruzhin, son of Ivan Shisman and titular "Emperor of the Bulgarians and the Romans", and sometimes ruler of parts of Bulgaria, participates in the crusade, the last time he is mentioned.
1453 AD: the Ottoman Turks under Mehmet II capture Constantinople, capital of the eastern Roman empire. Emperor Constiantine XI is killed in battle. At this point Holy Roman Emperor Frederick III should have proclaimed himself Eastern Roman Emperor to reunite the two Roman realms.
1460 AD: The Morea, the last part of the eastern Roman Empire that had been ruled by Constantine XI, is captured by the Ottoman Turks under Mehmet II. Despot Thomas Palaiologos flees to Italy and is recognized as titular "Byzantine" emperor by western Europe.
1461 AD: Trebizond is captured by the Ottoman Turks under Mehmet II, ending the Roman Empire there.
1465 AD: Thomas Palaiologos dies and his son Andreas Palaiologos is recognized as titular emperor.
1475 AD: Ottoman Turks under Mehmet II conquer Principality of Theodoro in the Crimea, a fragment of the eastern Roman Empire.
1502 AD: Titular Emperor Andreas Palaiologos dies, apparently without any known legitimate children. At this point the closest heir to the Palaiologos dynasty, Carlo III Tocco, should have proclaimed himself the titular emperor and/or Holy Roman Emperor Maximilian I should have proclaimed himself Eastern Roman Emperor.
1527 AD: The Duchy of Naxos, last fief of the Latin Empire, becomes an Ottoman vassal state, perhaps marking the end of the Latin Empire.
1579 AD: The Duchy of Naxos, last former fief of the Latin Empire, is annexed by the Ottomans, perhaps marking the end of the Latin Empire.>
1797 AD: The Republic of Venice, formerly part of the eastern Roman Empire, is conquered by France and granted to Austria in compensation for Austrian lands annexed by France. For centuries Venice had ruled various regions formerly part of the eastern Roman or "Byzantine" empire, at one point claiming to rule 3/8 of the Roman Empire.
1806 AD: Holy Roman Emperor Francis II, under threat from Napoleon, abdicates and declares the Holy Roman Empire dissolved.
So this brief list gives a few of the dates when the Roman Empire could be considered to have fallen.