The physical effects on the city and population of Constantinople upon its conquest in 1453 are well known: Sultan Mehmed II took the city by force, which meant that his army had free reign to loot and pillage. This would traditionally last for three days, but the sultan ordered his army to stop after only one. He was already planning on transforming Constantinople into his new imperial capital, and wanted to limit the damage the army would cause. With few exceptions, the whole population (which at this time amounted to less than 50,000 people) was enslaved, so repopulating the city became his first order of business. Canonically, one-fifth of all loot, including slaves, accrued to the ruler. Mehmed II resettled his fifth of the former population of the city along the shores of the Golden Horn and set them to work rebuilding, eventually enabling them to ransom themselves. He also, famously, set about forcibly resettling his subjects from other parts of the empire, whether Muslims, Greeks, or Jews, in the city in order to revive its economic life.

The relationship of the Byzantine elite with the Ottoman conquerors was very different. Mehmed II himself had a keen interest in Byzantine culture, and may have himself known Greek. He surrounded himself with members of the former Byzantine aristocracy, and it is logical that he should have done so - they could act as effective and reliable middlemen in the running of the empire. Shortly after the conquest, he identified the monk Gennadios Scholarios (who had sided with anti-Catholic factions prior to the conquest) among the enslaved, freed him, and invested him as Patriarch of the Orthodox Church. Kritovoulos of Imbros, a Greek in Ottoman service, describes their relationship as follows:

When the Sultan saw him, and had in a short time had proofs of his wisdom and prudence and virtue and also of his power as a speaker and of his religious character, he was greatly impressed by him, and held him in great honor and respect, and gave him the right to come to him at any time, and honored him with liberty and conversation. He enjoyed his various talks with him and his replies, and he loaded him with noble and costly gifts.

Mehmed II had the goal of winning over Orthodox elites and acquiring their full support in future conflicts against the Catholic states facing the Ottomans to the north (Hungary) and west (Venice et al.). But perhaps more importantly, he also had the goal of integrating the Byzantine world into his empire. It would be much easier to rule over the Balkans (and other Byzantine outposts like Trebizond) if he could have the support of their elite figures. Many elite Byzantines were brought into the Ottoman government, especially those who were willing to convert to Islam. The most famous example is probably Mahmud Pasha Angelović, on-and-off grand vizier between 1453 and his death in 1474. He was of joint Byzantine-Serbian origin, and had relatives still ruling in Serbia, whom he helped the Ottomans to win over and incorporate into the empire. Another was Mesih Pasha, the nephew of the last Byzantine emperor Constantine XI - had history gone differently, he might have become emperor himself, but instead he became an Ottoman commander and eventually grand vizier.

Opportunities for integration into the Ottoman system weren't limited to those who converted to Islam. While only Muslims could become high-ranking administrators or military commanders, Christian elites could become tax farmers, entrepreneurs, and financiers. Under Mehmed II, the customs administration of Istanbul was often contracted out to Greeks, some of whom were members of former Byzantine aristocratic families like the Palaiologoi, Kantakouzenoi, and Chalkokondyli. After the conquest of Serbia, some of these same families later administered its valuable silver mines, in effect profiting from Ottoman expansion. These families stuck around for a long time: as late as the 1570s we find a figure named "Michael Kantakouzenos" as a prominent tax farmer, merchant, and entrepreneur constructing ships for the Ottoman navy (admittedly, he may not have been an actual descendant of the medieval Kantakouzenoi, we don't know for sure).

All this is so say that many former Byzantines actually benefited from the newly-established Ottoman system. It wasn't just a matter of toleration - some elite Byzantines had the ability to adapt to and thrive within imperial space established by the Ottomans. Recognizing this goes a long way toward explaining the nature of Ottoman success, which depended on winning the cooperation and support from a wide enough spectrum of conquered peoples to make Ottoman rule stable and lasting.

These topics are covered in a recent survey by Molly Greene called The Edinburgh History of the Greeks, 1453-1768 (2015), which is a great starting point for the history of Greece under Ottoman rule. On the intersection of the Church and the economy there's also recent Render unto the Sultan: Power, Authority, and the Greek Orthodox Church in the early Ottoman Centuries (2015) by Tom Papademetriou. I'd also recommend Halil İnalcık's articles "The Status of the Greek Orthodox Patriarch under the Ottomans" (1991) and "Greeks in the Ottoman Economy and Finances, 1453-1500" (1993), both available in a collected volume called Essays in Ottoman History (1998). For a general take on early Ottoman history that emphasizes the Ottomans' capacity to integrate conquered peoples, there's Heath Lowry's The Nature of the Early Ottoman State (2003).