Who has the remains of Tutankhamen and the relics of his tomb—is it in an appropriate place?
The mummy of Tutankhamen (also known as Tutankhaten and "King Tut") rests in a climate controlled case in his tomb, in a climate controlled box. This box allows tourists to see the mummy of the "boy king" (who died when he was eighteen or nineteen).
...the linen-wrapped mummy was removed from its golden sarcophagus [and placed in a] climate-controlled glass box. The case was designed to prevent the heightened rate of decomposition caused by the humidity and warmth from tourists visiting the tomb.
The tomb is located in the Valley of the Kings in Egypt. The mummy and the artifacts discovered in 1922 by Howard Carter created quite a stir throughout the world—not just in archaeological circles. Because Tut's tomb and the artifacts it contained were so well preserved, this pharaoh of Egypt has realized greater recognition after his death than during his lifetime. While his mummy remains in his tomb, many of the artifacts discovered buried with his mummifed corpse have traveled around the world several times, starting in the 1970s and again with the turn of the century.
Tut and the artifacts belong to Egypt. They are national treasures. The iconic mask of Tutankhaman's mummy remains in Egypt and will never travel again because it is too fragile. It would seem that the mummy is given a great deal of care and respect: without the climate controlled environment in which it rests, the mummy would deteriorate with time, and nothing of the young pharoah would remain but the contents of his tomb. In this way the world has been able to learn a great deal about Egytian history, familial relationships in Tut's era (especially about Tut's parents), his cause of death, and a way of life that had been (in the past) surrounded by mystery.
If you are speaking of his physical remains, they are largely kept with his mummy in the Cairo Museum of the Antiquities. A number of artifacts, including his mummy case have been displayed in the United States and London.
The condition of his physical remains indicates some treatment improper for a monarch of Egypt. It appears that after its discovery, it was not properly re-wrapped, in fact the discoverer claimed never to have unwrapped the mummy in the first place. The arms, hands and legs of the mummy were amputated to allow bracelets to be removed. The head was removed to allow it to be removed from the mummy casing. The archaeologists responsible for this damage went to great lengths to disguise what they had done. Pictures taken in 1978 indicate that Tutankhamen's right ear and even his penis had been removed from the mummy although they were present in 1926. There is some evidence of carbonation of the remains, and unless steps have been taken recently to preserve it, the deterioration is likely to continue.