The wikipedia article about the Byzantine emperor Heraclius states, with a reference to Davis, Leo Donald. The first seven ecumenical councils (325–787): their history and theology (1990 ed.). Liturgical Press, that "one of the most important legacies of Heraclius was changing the official language of the Empire from Latin to Greek in 620". Obviously, a large part of the Eastern Roman Empire constituted of lands populated by Greeks, but what prompted this change in exactly this period? What made Latin less powerful?
Greek was always lingua franca in the east, before Romans even conquered the region. Romans already translated all official documents to Greek in the east before the split. And the transition from Latin to Greek as "official" language of the Byzantines was not a singular event, but a centuries long process. By the time the Turks were knocking at the gate there were corners of Byzantine bureaucracy that were still writing in Latin, and military orders were still given in Latin due to the disparate amount of languages spoken by the laypeople. But to directly answer your question, nobody in the Byzantine empire really even spole Latin anymore, and Greek was culturally dominant for centuries already