Assuming you mean emperor Constantine XI, unfortunately no one has any idea where he is buried, or if he was even buried at all.

He definitely joined in the fighting at the end, possibly dressed as a regular soldier so he was unrecognizable. But

“no eyewitness author whose work still survives was anywhere near the emperor at this crucial moment. All members of the imperial retinue were slain and there were no survivors to provide accurate reports.” (Philippides and Hanak, p. 232)

Some accounts report that he hanged himself, or that he stabbed himself with his own sword, or he asked his own men to kill him, or simply that he was one of the multitude of soldiers killed in battle with the Ottomans. The accounts that say he was killed by the Ottomans also suggest he was injured, then got back up and kept fighting until he was killed, and then his head was cut off and maybe carried around by the Ottoman troops on a lance. Some sources say the Ottomans looked for his body and couldn’t find it. One report says that they found a body with clothing and armour that had imperial symbols on them, so they assumed it was the emperor, but no one was really sure.

In short, none of the surviving sources from the siege actually had any idea what happened to the emperor, his body and/or his head.

No contemporary sources mention any burial site, and the Christians who were still living there in later decades had not identified any specific site. But legends did spring up, and by the 19th century there were several supposed burial places. Some believed that Constantine was buried in the yard of an inn in the Vefa neighbourhood, which may have been where prisoners were executed in the weeks after the siege. Another traditional site was the church of St. Theodosia, which is identified with Gül Mosque today, but it might not be the same place. Other suggested burial sites include the Church of the Holy Apostles (later torn down, now the site of Fatih Mosque), and that his severed head was buried at Hagia Sophia.

Mehmed was interested in finding his body because it would be a great propaganda victory in addition to the actual conquest. But the Ottomans probably couldn't find it either, so he simply declared that one of the bodies (the one with the imperial insignia) was the emperor. Then the Greeks would not have any reason to believe the emperor had escaped and would be coming back to rescue them. There were, however, stories that the emperor was actually “sleeping” and would re-emerge at some point and take the city back. This is just another variation of the "king under the mountain" folktale though, which is known all throughout Europe and Asia.

Presumably these legends and conflicting claims about burial sites wouldn't have happened if they had actually found his body and buried it.

“The grave of the last emperor has never been discovered and it is unlikely given the extensive rebuilding in Istanbul that the site will be soon discovered.” (Philippides and Hanak, pg. 564)

So you could go visit numerous sites which claim to be the burial place, but if he was ever buried anywhere, it was probably just in a mass grave with other unknown Byzantine soldiers.

Sources:

Marios Philippides and Walter K. Hanak, The Siege and the Fall of Constantinople in 1453: Historiography, Topography, and Military Studies (Ashgate, 2011 - particularly the "myths and legends" section, pp. 231-288)

Steven Runciman, The Fall of Constantinople, 1453 (Cambridge University Press, 1965, repr. Canto, 1990)

Jonathan Harris, The End of Byzantium (Yale University Press, 2012)

Edited to add another source:

Donald M. Nicol, The Immortal Emperor: The Life and Legend of Constantine Palaiologos, Last Emperor of the Romans (Cambridge University Press, 1992)