How to remove wall mounted mirrors without damaging wall?

Pulling the mirrors with suction caps can help you to a great extend. Else you can try with windshield removal wire. Start at one corner and then work on it down the length of the mirror, between the adhesive and the wall. In some cases, you will need few drywall works to clean it up.

If it is an old pier glass, there were two models : if it rests on a larger piece of wood it is mainly fixed at the basis with one peg at the top, else it is completely pegged and difficult to remove. In both cases you have to spoil the wall to remove it.

It depends how the mirror was attached to the wall. If it was merely screwed, then there should be no problem. If it was glued using tile or mirror adhesive, then there is very likely no way that you can remove it without damaging either the mirror or the plaster to which it's affixed. I'm afraid that you almost certainly have to say goodbye to your mirror - the only way to remove it will be with a hammer and chisel - and will have to make good the wall once you have removed the mirror and adhesive.