Some of the Emperors were buried in the Church of the Apostles.
The Church of the Apostles was built for the objective, to hold relics of all the apostles in that church, and to also hold the remains of Emperor Constantine (likewise known as "Equal to the Apostles) If I can remember, they could only get three Apostles, and later on other Emperors were also buried there.
I do not believe one can find many remains of any Emperors or Patriarchs in Constantinople. First, there are mentions of looting the tombs of both Emperor and Patriarch in the fourth crusade (among other things, many ancient sculptures of bronze ect melted down to make coins). I do not know where the remains of Emperor Constantine went through history But it is clear, that the remains of Emperor Justinian was included in the looting in the fourth crusade as his tomb was mentioned specifically.
By then, I do not believe many remains were left. Next, there was the Fall of Constantinople, where again anything remaining were looted, and later those churches destroyed and mosques built on top of them.
I feel sorry for any Christian buried in Constantinople. Then again I feel sorry for anyone buried in any, now major, city. (such as Paris for example...)
here are some pictures of results of excavations in Turkeyhttp://www.haber365.com/Galeri/Marmarayda_Bulunan_Tarihi_Eserler/1/
Notice all those skeletons... I feel very bad for those people buried there. I wish I knew who they were. They were people
This is the one that strikes me the most:
About the Church of the Apostles, there have been some excavations around the mosque, but they have ceased. It is known, that there is a Byzantine Chamber found under the mosque over the old site found during renovations. It was boarded up, and allowance to get in was not granted to archeologists.
I read that archaeology in Turkey is priced so only really Turks can conduct excavation. Foreign archaeological teams are put for such a price no one can pay for it. It is very unfortunate, as also the government does not seem to have much care for the archaeological value in Constantinople.