I've heard various arguments that the Byzantines were dealt the mortal wound at Manzikert in 1071 which allowed the Turks to claim most of Anatolia and set the stage for the later sacking of Constantinople by the Latin Crusaders. I've also heard that the it was the sacking itself that set the Empire on a course to its ultimate end. With the fall of Constantinople in 1453 being generally accepted as the end of the Byzantine Empire proper; when did the slope toward that end begin?
The fourth crusade was the turning point. The crusade was high-jacked by Venice to take revenge on the Byzantines for past deeds: imprisonments, break of contract, etc... The crusade was aimed to land in Egypt originally, as it was seen as the main threat to taking Jerusalem back. However, since the crusaders could not pay for the large Venetian feet, it was arranged that they would do a few missions for Venice first. Not all crusaders agreed, but the majority saw no choice. Even the Pope was unhappy with this state of affairs.
So, once Venice took control of Constantinople, a series of civil wars and coup happened. This weakened Byzantium to the point where it could not recover. From there on, it was just a matter of time before another power took control.
As a side note, the Byzantine fleet was in great part responsible for the might of the empire. Once the Empire gave this to Venice to build and use, it was just a matter of time before the fleet degraded beyond the local ability to rebuild it. An example of why out-sourcing is bad.
I don't think it is possible to idenitify a single point in history as beginning the "slope toward the end". Such thinking results from the simplistic model of an empire's history as consisting of two segements: "growth" and "decline". In reality, the history of the Byzantine empire is a complex sequence of alternating growth and decline.
I'd say that the first high point of the empire was the end of Justinian's rule, when the borders of the Byzantine empire bore some resemblance to the old Roman empire. A large numbers of events weakened the empire from this high point. They include the loss of much of the Italy to the Lombards in the 6th century and the gradual loss of Levant, Mesopotamia and North Africa to Muslim Arabs starting from the rise of the Umayyad Caliphate in the 7th century.
In 1054 the Byzantine Church formally split with the Church of Rome, after a long period of growing tensions. The cause of the tensions was a combination of political rivalry between the Byzantine Emperor and the Pope and religious divergence due to movements such as Monophysitism, Monothelitism and Iconoclasm within the Byzantine Empire. This eventually contributed to the demise of the Empire due to its religious and therefore political isolation. Territorial decline continued with the loss of of remaining Italy to the Normans in the 11th and 12th century and the gradual Seljuk takeover of Anatolia starting in the 11th century.
Another high point was the Komnenian restoration in the 12th century, during which much of Anatolia was temporarily recovered. After that, there was return to severe dynastic strife which was another major factor in Byzantine decline.
During the Fourth Crusade, Constantinople was taken by the (Roman Catholic) crusaders in 1203. The key role in this event belonged to the naval–commercial rivalry between the Byzantine Empire the Republic of Venice. The crusaders couldn't pay for the fleet Venice supplied them and agreed to attack Constantinople as a compensation. The fall of Constantinople caused the split of the Empire into three parts: Empire of Nicaea, Empire of Trebizond and the Despotate of Epirus. The crusaders created their own state: the Latin Empire.
The Empire of Nicaea under Michael VIII Palaiologos managed to restore the Byzantine Empire by taking back Constantinople in 1261. This caused the demise of the Latin Empire, in what is probably the last high point in Byzantine history. The rise of the Ottoman Empire in 1299 caused the gradual eclipse of the Byzantine, with Constantinople falling again in 1453 and remaining "mainstream" Byzantine territory in Morea (Peloponnese peninsula) falling in 1460. The Empire of Trebizond fell in 1461 and the Despotate of Epirus lingered on until 1479 when it also was taken over by the Ottoman Turks.
A prior article mentions the empire of Justinian (and Leo, by extension), but I would argue that these are 'Roman' empires which are terminated by the eruption of Islam over much of the East Roman Empire.
This was a pretty traumatic event which led to some serious results. Among them, the abandonment of Latin, abandonment (with some exceptions) of universal pretensions in the western Mediterranean, acceptance of defence against the Muslim and recovery of the Holy Places.
And then there was iconoclasm on the cultural front. If you look carefully, there is a discernible shift in Weltanschauing in the 8th century in the lands ruled from Byzantium which may be called the birth of the Byzantine Empire and civilization and civilizing efforts through the Balkans and beyond Now, I am aware that my comments mimic those of Arnold Toynbee, but this is one case where his old thesis does strike true.
Of course, the Byzantine state enjoyed considerable success under generals like Nicephorus Phocas and Johm Tzimisces and others long before the time of Basil II (Bulgaroctes) But was it already becoming a feudal state? Surviving records from the maritime Themes show central control, but what was happening in the 'wild west'? Doesn't the epic poem Digines Akrites portray a feudal society on the frontier?
Anyway, I reckon that a feudal army went to Manzikert in 1079 and the rot set in. The Commneni had a very faint chance a hundred years later. Byzantium was still the strongest regional power at the time of the first crusade. But feudalism, both Byzantine and western, prevented any combined effort. Then the battle of Myriokephalon (various spellings) destroyed forever any Byzantine pretensions to power forever.
The empire limped on a for a few more centuries and the civilization may have a few sparks of life left, but I nominate Myriokefalum as the death knell of Byzantiu
From what I recall from historical texts that I've read in my travels, the event that seems to have started the irreversible decline was the Fourth Crusade, when instead of heading to free Jerusalem, the crusading armies attacked and sacked Constantinople. With large parts of the Empire fragmented into Latin states by the armies that had attacked them, it helped the Seljuk Turks (and later the Ottomans) hold the Anatolia area of Asia Minor and strengthened their position for furthur conquests as those states fell.
References: Wikipedia's article on the Fourth Crusade
Personally, for me, the turning point was Manzikert. It wiped out a good portion of the fighting men of the empire, and caused the Seljuks to take the eastern part of Asia Minor, which was a large source of manpower for the emperors under the theme system. So with the threat of a Seljuk invasion, Empire responded with a plea for help to the West, launching the Crusades. The final blow to the Byzantine dominance was the sack of Constantinople by the Fourth Crusade, from which the Empire never really recovered.
The thing that really started the fatal downward skid for the Byzantine Empire was the arrival of the Turks from central Asia. They were the best warriors in the area, and just as importantly, were dedicated pastoralists. The best land in the area for their purposes was in central Anatolia, which just happened to be smack dab in the middle of the Byzantine Empire.
For the Byzantines the wealthy coastal areas were more important, but whoever held that central anatolian plain could attack any point on the coast at will. Once the Turks showed up, nearly every military effort of theirs (that wasn't spent on internectine warfare) was spent trying to get that good grazing land in the heart of the empire. The Byzantines could slow them down, sometimes even stop them, but they couldn't turn the situation around.
The Crusades started out as a desparate attempt by the Byzantines to get some help kicking the Turks out of anatolia. Some of these worked better than others. As mentioned by others, the Fourth Crusade backfired completely. But it still took a couple of hundred years (and the invention of cannon) before the Turks finally took Constantinople and extinguished the empire for good.
i would date the "irreversibility" of Turkish power to the Sack of Constantinople in 1204. This resulted in the division of the former Byzantine Empire into the Latin Empire, the Kingdom of Nicea, and several "splinter" groups (e.g. Epirus and Trebizond), with Constantinople a "football" between the first two. Without access to "Latin" mercentaries, the ultimate "winner," Nicea, could field armies only a fraction of the size of Byzantine armies a century earlier.
It's true, as T.E.D. pointed out that the Turks became a threat to the Byzantine Empire by the mid-11th century. Yet the rise of the Komnenos Dynasty, 1081-1185 involved A "rollback" (or "reversal") of Turkish power out of much of western Turkey during thaT period.