I'm not sure if there's a consensus, but I can go over some of the big dates and why they make useful demarcations.

Dedication of Constantinople, 330. Diocletian had previously divided the empire, but Constantine created an equal counterweight to Rome in the East. You can't really go earlier than this date.

Death of Theodosius, 395. Briefly before his death, Theodosius had united the empire under one ruler for the last time. After this point, the administrative division was permanent and there would always be a separate Western and Eastern Emperor.

Deposition of Romulus Augustus, 476. The traditional end of the Western Empire. He was the son of Orestes, a Germanic general under Julius Nepos (see below), and a Roman woman so he could become Emperor whereas a full-blooded German like Orestes would not be palatable. His father deposed Julius Nepos in 475 only to have his son deposed by another German, Odoacer. Odoacer felt it was politically expedient to merely rule in the name of Julius Nepos and the Eastern Emperor. As the East was undergoing a civil war at the time, the point was not pressed.

Death of Julius Nepos, 480. The pedant's end of the Western Empire. Nepos was originally appointed Western Emperor by the Eastern Emperor Leo I and dispatched to overthrow a puppet emperor of a Germanic general (seeing a theme?) in 474. After he was ousted by Orestes, Nepos fled to his power base in Dalmatia (modern Croatia) and survived a few more years. Odoacer paid lip service to him for a time, but probably conspired to have him murdered and he was assassinated by his soldiers in 480.

Reign of Heraclius, 610-642. Probably the latest you can go to find a convenient dividing line. During his rule, many enormously consequential events happened. The language of administration became Greek versus Latin, the Empire permanently lost Egypt and the Levant to the Arabs, and (under either him or his successors) the reformation of the classical Roman provinces into military districts called themes. From this point on, the Empire would more or less remain the familiar predominantly Greek state confined to the Balkans and Anatolia of medieval times versus a Latin-speaking Mediterranean power.

EDIT: It's also worth pointing out that the Byzantine Emperor was seen by Westerners (again, more or less) as "the" emperor up until 800, and the excuse that the Pope used in crowning Charlemagne emperor was that the throne was vacant (i.e. occupied by a woman, Irene). Thereafter, the west increasingly belittled the "Greek Empire" and the Byzantines responded by imprisoning any diplomats accredited to the "Emperor of the Greeks."