I am assuming these are glued-on, not mounted with hidden slip-in racks or tray fasteners ? If not glued, no problem whatever to remove them (at least for a glass/mirror man).
Several ways I have run across that people have handled these - some people use a mixture of several in one room, usually in a symmetric manner:
1) find a mirror sales/installation company who has a mirror saw (assuming these are surface, not flush mounted) - they can cut through the glue in no time flat, leaving just a small amount of glue to be picked off and a bit of drywall scraping damage to be patched with drywall compound and repainted. The saw is like a bandsaw with a diamond or carbide-coated wire blade like that used to cut granite but on a small scale, that cuts through the glue behind the mirror. Can be done by machine or by hand with 2 people. There are also razor-thin very long saw blades to do this. Probably about $20-30/mirror if you have multiple mirrors.
2) build or have a finish carpenter make finished wood frames to fit around them, and fill the frames with tapestries, paintings, wood veneer, posters, or whatever.
3) above, but deeper with shelves as a curio or display or memorabilia or hobby display case with the mirror showing through behind, or cover mirror with stick-on paper or veneer first
4) hang cloth on it, tightly stretch and fasten around the edges, then pin or hot glue knickknacks or photos, etc on it - I saw one stunning one with special doll hangers to suspend antique dolls, with a clear acrylic plastic case cover built over the front to protect it from dust and such
5) hang tapestries or persian rugs or scene/animal area rugs over them
6) paint a scene on it with oil paints or acrylics
7) apply those stick-on scenery pictures designed for kids rooms or to go over panelling - essentially contact-glue backed scenery wallpaper
8) cover with fancy art paper - there are some incredibly fancy and intricate interior design papers - I would think photomount spray adhesive or spray contact cement on both surfaces per instructions would work fine
9) have shrink-wrapped with a design or photo - the kind of shrink appliques they are putting on buses and vans for advertising - getting relatively cheap with new high-volume printers at sign shops, and available in a LOT of patterns and photo images
10) cover with a double-layer acrylic hinged face that you can open up (hinged at bottom) and open back acrylic piece, and lay matted photos pressed between the two pieces, then close back up and hinge backup into place - makes a dynamic display case for photos, diplomas, certificates, etc that can be changed at will
11) just conceal with highboys or bookshelves, with or without a drape to conceal the mirrors behind
12) applique with peel and stick window frosting or stained glass effect sheeting, or have acid etched in a pattern (or learn to do it yourself)