How is the mirror attached to the wall? It it held on with a frame that can be unscrewed or is it glued to the wall? If glued you could try sliding a stiff putty knife behind there to attempt to pry it off. The odds of getting it off in this case without snapping the glass can be pretty slim.
Can't say I have tried either of these methods, but I trust Popular Mechanics. They generally know what they are talking about:
Cutting out the drywall seems pretty foolproof if you want to save the mirror. Of course, it will take some patching, but people often give away scraps of drywall on Craigslist and such. Those patches should be just the thing for replacing the spot. Just keep in mind that a bathroom should have "greenboard," which is moisture-resistant drywall.
Most mirrors are installed using an asphalt type of adhesive, some are contractors adhesive.
in any case if the mirror is really old it is going to be fragile. Lots of painters tape or duct tape should be used to keep it as best you can intact should you crack it when removing it.
Piano wire is often used as its very strong and when used in a sawing motion will cut through the glue with ease.
You need to slightly apply outward pressure with a wide putty knife or wood shims but not to much or it will crack. Then using the wire attached to vice grip pliers saw back and fourth until it beings to loosen. You need a few sets of hands for this. This method works well if you have access to both sides of the mirror.
Another method that folks have used is simply water. The wall material behind the mirror will need to be replaced anyway as you will not be able to remove the glue on the surface of the wall board. So.....place several towels on the bottom of the mirror to catch extra water, then using a spray bottle start applying water behind the mirror. As you do this the unpainted wall board, hopefully will begin to soften up and come apart. You again simply by carefully applying some pressure with putty knife, or some wooden shims purchased for hanging doors keep the pressure on the glass and keep wetting the wall behind it. This takes a bit of time to succeed but you will get the mirror off this way. Of course masking tape must be used for safety.
Once mirror is off, do not lay it on side and carry it. Keep it upright, wear heavy gloves, pants and long sleeve shirts. A few drop cloths should be used in case the mirror breaks so you do not get any glass shards in the carpets or around the room. Take your time and you will be successful with this.