With the conquest, the Roman Empire was officially ended, and at the time, it was still seen as the Roman Empire, as the Byzantine Empire is the name used mainly by modern historians. I'm curious how the inhabitants of the city that birthed the empire reacted to the news that it had been destroyed. Was there mourning, ambivalence, despair, apathy, anger...?
Constantinople fell on May 29, 1453. The news reached Venice on June 29, during a meeting of the Great Council. A message was promptly drafted and sent by courier to Pope Nicholas V. It reached Bologna on July 4, and Rome on July 8.
When the Fall of Constantinople was announced to the Roman people, the initial reaction was a combination of horror and disbelief. Everyone knew that Constantinople was in terrible danger; but the city loomed so large in the popular imagination that its conquest seemed impossible. In the weeks between the advent of the Venetian message and the arrival of corroborating reports, rumors circulated wildly. Some claimed that the city had not fallen, could never fall. Others said that it had been seized by the Turks, but then swiftly recaptured by Christians. Others still warned that the Sultan, having conquered the New Rome, was already on his way (with a large Turkish fleet) to subjugate the old.
For Pope Nicholas, the news was both a crushing blow and a desperate inspiration. Constantinople, Nicholas resolved, had to be retaken; and to that end he sent couriers to all the major Italian states, ordering them to cease their wars and commit to a common cause against the Turks. A crusade was formally proclaimed by papal bull at the end of September. Yet despite a brief surge of enthusiasm (and a still briefer bout of fund-raising), these efforts came to nothing.
Back in Rome, other reports had by now confirmed the truth of the Venetian missive: Constantinople was in the hands of the Turks, and the Sultan was contemplating an invasion of Italy (that threatened invasion, as it happened, only took place in 1480). Roman humanists turned their fertile pens to laments - and, somewhat later, to elaborations of the curious theory that the Turks were actually distant descendants of the Trojans, and that their conquest of Constantinople represented a long-deferred revenge on the Greeks.
[I am working well outside my usual area of expertise here. Most of this answer was gratefully cribbed from the second volume of Ludwig Pastor's vast and vastly scrupulous History of the Popes, conveniently available online in all of its compendious glory.]