How do I prepare a wall for backsplash tile?
Assuming this is ceramic tile - not glass or stone - and you are going to do the standard mastic installation, if there is only mild damage (for example, a few 4-inch patches ripped off) either just ignore it or use joint compound to fill.

Joint compound is not strong at all.  It is pretty crumbly and soft - i.e. it's not plaster, even though builders nowadays use it to skim-coat walls for texture (you can easily gouge it with your fingernail on such installs).  So if there are large areas of damage, I would cut out the sections of drywall and replace them.  This is not difficult, because although you should still mud-and-tape the joints, you don't need to sand much since you'll be covering it with tile.

Final comment - I wish, on my installation, that we had added 1/4" hardiboard over the sheetrock (with const. adhesive and screws) to stiffen the wall.  My wall has the garage door in it, so the wall gets flexed and the grout cracks.  Just FYI.

(Adding Fein Multimaster picture, see comment below)