Wow - that's hard to cover in a short reddit post! All in all, there was a large influence in many aspects of everyday life and administration, but it's not always easily discernible - it basically depends on how much you want to see them. Similarities can be due to direct borrowings from the Byzantines, or indirect borrowings through the Arabs, Persians and Venetians, who interacted with the Byzantines for centuries. There are also things that the Arabs & Persians transmitted to the Byzantines and then to the Ottomans. Many Byzantines (and "Post-Byzantines", i.e. Christian subjects during the 15th and 16th century) and later Phanariotes entered the Ottoman military and administration, both as Christians and as converts, bringing along their know-how in law, diplomacy, naval warfare, taxation etc. . All those influences were mutated/disguised/transformed or buried among the already existing heavy Persian and Arab influences on Ottoman culture. The Ottomans wanted to be seen as legitimate rulers of their christian byzantine subjects (hence the title Kayser-i-Rum, i.e. Caesar of the Romans), to stabilise the newly conquered territories, while at the same time retaining their islamic-Turkish credentials to avoid opposition from the Ottoman aristocracy and clergy. In that sense, any direct borrowing from the infidels would seem suspicious and was accordingly disguised. Add to that the fact that we don't know very much about byzantine practices in many things, so we can't pinpoint a direct byzantine influence in e.g. Ottoman taxation, even when it's staring us in the face. That's because very little survives of the once huge amount of byzantine official documents.

In short: a) Ottoman Turkish had a number of greek words, and latin/italian ones that passed through byzantine Greek, esp. words for seafood, local plants and animals, and naval terms.

b) Other examples of direct appropriation: architecture (most mosques are reinterpretations the Hagia Sophia's dome-on-square plan or imitations of Sinan's works, that were also inspired by the Hagia Sophia, to the extent that they weren't converted churches, and many byzantine palaces and monasteried were appropriated and repurposed), music (partly), musical instruments (partly), foods (partly), medicine, traditions and beliefs such as the evil eye, and some aspects of administration, taxation, law and land survey, especially regarding their Christian subjects. The Orthodox church, which played a crucial role in the administration and justice system of Orthodox Christian subjects, upheld many byzantine traditions. Since the Turks advanced gradually through Asia Minor towards Constantinople in a few hundred years, they had plenty of time to appropriate everything from farming methods to land administration, and then rename and mutate it beyond recognition.

c) The question of "how much of Ottoman civilization can be interpreted as a continuation of Byzantine traditions under Islamic-Turkish rule" is hotly debated: it depends on each person's ideological background and their knowledge of both Ottoman and Byzantine history, as well as their general viewpoint. A concrete example is the Ottoman timar system, i.e. land grants given by the Sultan to members of the military class as compensation for their services. Was it an adaptation of the byzantine pronoia system? We don't know. Another example was the byzantine stremma (about 900 sqm) which became the ottoman dönüm (900-1000 sqm) - possibly. Was the ottoman cadaster system and the taxation of land influenced by byzantine and venetian practices ? again, we can't tell for sure. Byzantine legal historians like to point out that the Ottomans appropriated several aspects of late byzantine administration, without going into deep details, and post-byzantine legal historians discuss the similarities and differences between the two systems, but take them as they are, without trying to establish if the ottoman practice originated in the byzantine one or not. The subject is huge, and even if it's broken down into smaller subjects, ther is no singular, concrete answer.