The tomb itself is administered by Greek Orthodox, but is open to all believers. The priest who was overseeing the tomb when I was there, as visitors could enter and leave the tiny space only one at a time, looked to be almost a hundred years old. Entering the space made the hair on my neck stand up and is one of the most vivid memories of my life.
Also the Catholic Church and two other churches as well, not just the Greek Orthodox take care of that shrine.
What brought the needed repair from what I had read was a direct warning from the Israeli government that the Edicule was in danger of collasping. That was reason for its repair.