In order to talk about Venice in relations to the Byzantine Empire, the city of Venice was indeed once a part of the Eastern Roman Empire. In the beginning, the introduction of the Republic of Venice into the Byzantine Empire was the genius of Emperor Basil the Bulgar-Slayer. Finding the growing threat of Tsar Samuel who was taking major cities in Dalmatia and cities on the coast of the Adriatic, Basil did not like the idea of having to take his eyes off the East, and thus came up with a policy to keep his lands in the west, while not having to defend them himself. Basil used the Republic of Venice and at the time Doge Pietro Orseolo II, by granting them suzerainty of the Byzantine Dalmatian Coast. The trade off was that Venice gained the valuable Dalmatian coast (where pirates were harboring to attack merchant ships that were sailing for the harbors of Venice) but in return had to defend the land from invasion and keep it in Byzantine hands.
The Republic of Venice and Constantinople had friendly relations until Emperors began to cut off the ties. Uptil then, the Republic of Venice was valiant in their support of Constantinople, even risking harsh finger pointings from the Holy See (the Vatican). In one case, the Republic of Venice joined Constantinople (Emperor Alexius Comnenus) in the destruction of a Norman fleet who was capturing cities along the coast of Greece on route to Constantinople itself (from Southern Italy).
But in the end, the Republic of Venice was more a plague to the Byzantine Empire then it was ever helpful by a long shot. We owe the destruction of Constantinople and quite possibly the destruction of the Empire to Venice, and it's greed for gold. The 4th Crusade sacked Constantinople of all it's art work and beauty in order to pay off Venice for it's transportation east.
We all know that these barbaric Crusaders in the end, established the brief Latin Empire in Constantinople, depriving the true line of Caesars for over 5 decades.
At the fall of Constantinople, the Republic of Venice did send aid to the city, but much to the city's embarressment, a good portion of the men sent sailed away before the seige actually began. It was not Venice but the men from Genoa (the arch rival of Venice at the time) that fought with the Romans till the very end.