This is a frequently-cited anecdote that, if not completely false (though I think it is), is at least extremely misleading. The idea that Byzantium on the eve of the Ottoman conquest was too preoccupied with splitting theological hairs is absolutely taken for granted and doesn't remotely reflect the broader historical and political context of the event.

It is true that the Byzantines had always been interested in theological niceties. St. Gregory of Nyssa (4th c.) gives us the famous anecdote that you couldn't ask for your change from a shopkeeper in Constantinople without also getting lectured on whether Christ was a created being or co-eternal with God. The polymath, historian, and monk Michael Psellos (11th c.) explained that angels don't really have wings, they're just painted with them because they have to ascend to get back to Heaven. (Incidentally, he also explains that they don't really have bodies - and thus, sex - since angels only look like people to show that they have intellect and reason.) Even after the Fourth Crusade (1204), there was a great deal of debate on the pros and cons of Western (Catholic) and Eastern (Orthodox) practice; certain later monks (e.g. Manuel Chrysoloras, who wrote the first Greek textbook used in the West since Antiquity) and emperors (e.g. John V, r. 1341-1391) ended up converting to Catholicism, though they were never really successful in getting the populace or even most of the clergy to follow suit. The Union of the Churches pursued by John VIII (r. 1425-48) and Constantine XI (1448-1453) seems to have mostly been an attempt to gain Western military support against the Turks rather than a sincere bid at reconciliation of the Schism of 1054. So in one sense, yes - there were great theological debates going on in Constantinople around the time of the Ottoman conquest. But these theological debates arose from a pragmatic cause: Should the empire become Catholic because it was correct? What about to attract Western defenders? Would it even matter if they did?

The heart of the problem lies in the assumption that these profound theological arguments were in any way incompatible with a competent military defense of the city. Constantinople had been besieged lots of times before this - by the Avars in 619, the Arabs in 717/8, the Rus' in 860 and again in 941, to name only a few instances - but these sieges had never been fatal, even though they were contemporary or near-contemporary with periods of serious religious tumult: Monophysitism/Monothelitism wasn't satisfactorily addressed until the Third Council of Constantinople (680-1), while Iconoclasm first shows up around 726 and isn't completely solved until 842. By the fifteenth century, the empire had certainly become quite weak - but this isn't because of some prolonged religious debate. The Fourth Crusade absolutely decimated whatever cohesion and stability the empire had managed to acquire over the previous couple of centuries, and even though the Palaiologoi eventually reconquered Constantinople the facts of ruling in the eastern Mediterranean were completely different. Some of the Palaiologan emperors were relatively competent rulers - John VI Kantakouzenos only ruled for eight years (1347-1354) but he managed to recover Thessalonike in 1350 from the Zealots who had held it since 1342, and even if John VII (April-September 1390) usurped the throne he then successfully defended Constantinople from the siege of Bayezid I (1399-1402) and ruled Thessalonike as a kind of independent kingdom - but by this point the empire was simply incapable of resisting the Turks for very long without serious outside assistance. The Crusade of Varna (1443-4) might have been able to provide this help, but Murad II crushed the Papal army and the West made no serious further attempt to assist Constantinople. Centuries of reliance on Venetian, Frankish, Catalan, Genoese, and even Turkish mercenary forces meant the Byzantine army and navy had stagnated. Without a competent defensive force in place, it was inevitable that Byzantium should be eventually overpowered.

By the time Mehmet laid siege to the city, Constantinople itself was practically the entire extent of the empire. At the same time, Mehmet held all the key strategic resources that had been indispensable to Byzantium in its heyday, as far back as the 10th or 11th centuries: Anatolia, the source of the empire's soldiers; Thrace, the source of its food; control of Aegean communication and trade routes; and so on. In that situation, I'd be debating theology, too. But ascribing the death blow of a 1500-year old empire to simple religious obsession in a moment of crisis is absurd. The Byzantines defended their city as best they could; it's no fault of their own that they were outmatched.

Religious debate over philosophical details was a basic feature of life in Byzantine Constantinople, that is true, but this anecdote is often told as a way of saying that the Romans just did not defend their city, which is false. Even nuns fought in the defense, throwing rocks off the northern walls when Mehmet managed to put some boats in the Golden Horn through Pera, as Jason Goodwin states in Lords of the Horizons: A History of the Ottoman Empire.