Is it appropriate to put a rosary on a relatives headstone and leave it there even if it starts to discolor and rust the headstone? Would it be a desecration of the grave if someone removed it because of possible negative effects?
I don't believe that there is any formal Catholic doctrine on the use of rosary beads (technically "a rosary" refers to the prayer, or rather sequence of prayers, not to the physical beads themselves). While the Catechism of the Catholic Church mentions the rosary (always referring to the prayers), it contains no description of how the beads ought to be treated. At most, it simply says in paragraph 1676:
If the beads are blessed, as they often are, then of course they ought to be treated with reverence.