How would it affect the Habsburg hegemony? Too much territory and little incentive to go colonizing, unlike the smaller western European countries that were seeking more lands. My theory: Probably not, just as the Ottomans didn't really become one. Greek expatriates contributed to it, but no doubt the Italians could have received Greek influences anyway. Would it affect the start of the Renaissance in Italy?
Though eventually someone would have sailed westwards anyway, all it would take is someone like Manuel Comnenus to deport the Venetians and block their trade. My theory: Possibly, if the western merchants were allowed to continue their eastern trade via the Empire, there would have been less incentive to find alternate routes. Would it affect the time of the discovery of the New World? Let's also assume that it is generally able to keep its border in eastern Anatolia regardless of later historical events (like Tamerlane's invasion).
Let's assume that the Byzantine Empire managed to survive with its AD 1025 borders, minus southern Italy/Sicily.
