I don't really know how much information historians have to offer, so here's what I found online (before asking the question) :
Most of the emperors and their family members were buried in their sarcophagi and in different mausoleums. It seems like we have very detailed accounts of the locations of those burial places.
Constantine I and many others were buried in the Church of the Holy Apostles, it was the main burying place, but became overcrowded with time.
During the sack of Constantinople, crusaders robbed the dead emperors of their valuables. Their imperial purple silk robes were snatched, unwoven and sold off. They are said to have found the body of Justinian I almost in perfect condition (not decomposed).
After 1453 the patriarch was transferred to the Church of the Holy Apostles from Hagia Sophia, but soon after he was made to leave that place too. The church was destroyed and Mehmet II built a mausoleum for himself there.
[This info comes from an article written by a russian orthodox historian, so yeah please fact check] : Ottomans destroyed all the tombs of the emperors and the only one to survive was the body of the Andronikos Palailogos, who died in a monastery after being overthrown and buried unceremoniously and secretly under the gateway of a small church, thus avoiding all the vandalism done by the Ottomans.
So yeah, that's all I was able to collect online and I hope someone experienced in Ottoman/Roman history would be able to fact check those and add other informations and present a fuller picture.
The question of the existence of any remains of bodies and tombs of Roman Emperors, that are known to us in the Modern Era, discovered by archaeologists, is equally interesting an charming. However, History tells us that we unfortunately, almost all of them have bee destroyed and lost to oblivion.
While the in the Roman Kingdom and Early Roman Republic the Romans usually buried their dead, later the practice of inhumation gradually decreased and the Romans started the use of cremation. Thus the bodies of the Roman Emperor’s were burned and their ashes were later preserved in imperial mausoleums. The Roman Emperor Augustus (27 BC- 14 AD) built the Mausoleum of Augustus. His ashes and those of other Emperors of the Julio-Claudian dynasty were kept there. In a similar manner others built later their own tombs for themselves and their families, such as Handrian, who built the Handrian Mausoleum. Lamentably, these tombs were later destroyed and defiled. During the Sack of Rome of 410 by Alaric, the Visigoths enthered the tombs and mausoleums of the Emperors, stole the urns and scattered the ashes.
We must also not forget the fact that there were also many bodies of Emperors, which were lost in the many wars of the Roman Empire, either civil of external. They were lost in the battlefield or were buried in unmarked graves. Also there is the example of Valerian, who was captured by the Persians, who was mummified and preserved as a trophy. Thus our search for the bodies of the Roman Emperors must go further.
By the end of the 2nd century AD, the Roman burial practices began to change and cremation slowly came out of use. With the continuous spread of Christianity throughout the state, the Romans started inhumating. Notably, when Constantine the Great moved the capital of the Roman Empire to New Rome (Constantinople) he founded the Church of the Holy Apostles, which was completed by Constantius II. The church was also a mausoleum, which kept both the bodies of the Roman Emperors and relics of the Apostles. There were buried all the Roman Emperors of Constantinople from Constantine I to Constantine VIII. The Emperor Basil II, refused to be buried there as he wished to be with his soldiers, in the Church of Saint John the Theologian, next to the barracks of the City. After this the Church of the Holy Apostles was full and the Roman Emperors were being buried in other churches and monasteries around the capital.
Unfortunately, the tombs of the Roman Emperors in the Church of the Holy Apostles in Constantinople were looted by the Crusaders during the Fourth Crusade of 1204 AD. They disturbed the bodies, damaging them and the tomb itself. As for the body of Basil II, after the Latins pillaged his tomb, they took his body and mockingly put a flute in his mouth, in order to ridicule him.The bodies were later completely destroyed with the Fall of Constantinople in 1453 to the Ottoman Turks. The Church of the Holy Apostles was demolished by the dervishes and the remains of the Roman Emperors were smashed, burned and completely destroyed.
Moreover, others would become monks in various monasteries, which meant that they would be buried there and later have their skeletons maintained in the monasteries ossuary. However this meant that the remains would be impossible to find if we discovered the ossuary, as they would be mixed with the bones of the other monks. However there hasn’t been such a discovery in Constantinople as most of the churches and the monasteries were turned into Mosques by the Ottoman Turks. It seems that with the passing of the centuries, the tombs of Roman Emperors which were well known were demolished and that the bodies were destroyed. Or those who were buried in private graves or in monasteries were forgotten and lost to History. Despite of all this, there is still a chance for the latter that they can be found.
A great example of this is the Monastery of Panagia Varnakova in Dorida, in Greece. This monastery was founded in 1077 AD by the Emperor Michael VII Doukas (1071 AD- 1078 AD). After the Fall of Constantinople in 1204 AD the monastery was under the rule of the Despots of Epirus, claimants of the Roman Throne. The Doukas-Komnene dynasty favoured the monastery and loved Panagia Varnakova so much that some of them decided to become monks there or to be buried in the entry of the church. Centuries later in 1919 AD the Greek archaeologist Anastasios Orlandos discovered the tombs of the Komnene Despots, claimants of the Roman Emperorship, in the floor of the church. Even today the tomb inscriptions can be seen, but alas, they have not been opened.
The only place where we can say that there has been a body is in Corfu, Greece. The body belongs actually not to a Roman Emperor, but to a Roman Empress which lived during the 9th century AD. This Empress is Theodora (815-867), the wife of the iconoclastic Roman Emperor Theophilos (805 AD- 842 AD). She was actually a Regent Empress, meaning that at one point she ruled the Roman Empire by herself after the death of her husband. She ruled for thirteen years (842 AD–855 AD) with the aid of an eunuch Theoktistos, as a regent for her son Michael III (840 AD-867 AD), who was only two at the time of his father’s death and thus unable to rule. Her body lies in a vault beneath the Metropolitan Church of Kerkyra.
