There are two ways by which you can fix the mirrors on the walls. One is with the help of screws and other one is through adhesive construction on its back side. Both the solutions equally fare well but in the second case the coating comes out when you remove the mirror. But if it is done properly then even a single scratch will not come. The modern tools are used so that the surface remains the same and there is no effect or marks on it.
You just have to take out the clips which can be done with the screwdriver. If you are not able to remove it by the above method then you can employ the wire and stretch it out with force. This will really work for you as pressure has to be exerted and the other person who can hold the mirror in position. Hence you can change the mirror without much trouble.