Because Christians couldn’t agree over who gets to “own” the holiest site in all of Christendom, the duty fell to a neutral party who are Muslims.
The Church of the Holy Sepulchre is reportedly built on the site where Jesus was crucified. There is a long history of fighting over this site.
“To this day, coexistence among the several Christian churches sharing the Holy Sepulchre is a delicate one. Catholic, Greek, Armenian, Coptic, Syriac, and Ethiopian Orthodox monks have resorted to fists more than once to defend their respective denomination’s rights and privileges in the church, as defined in an decree by the Ottoman Empire, known as the Status Quo of 1853.” (1)
Because of this, the fact that so many Christian factions have fought over it, there is a long standing tradition of having a neutral party, comprised of Muslims, keep the keys and let Christian priests and pilgrims into the site.