Cut large, straight cuts with an angle grinder, using a diamond blade attachment. Hold the grinder with both hands so that the blade of the grinder is perpendicular to the face of the tile. Apply pressure while you pull the trigger so that the blade eases its way into the face of the tile. Cut the perimeter, then cut diagonally across the square area to break the section up into pieces that are easier to pry away from the wall, or simply cut all the way through if you need to install something else in the tile's place.
Cut oddly shaped or intricate areas with your chosen Dremel tool. Use the oscillating grout-removal tool to work your way through grout joints if you are removing single pieces of tile, or use one of the diamond or carbide attachments on the rotary tool to cut circles, such as for mounting something on the wall or placing custom inserts into a tile wall.
Drill through the tile with a diamond or carbide cut-out drill bit for mounting things on the wall or cutting out for pipe fixtures, for example. A regular carbide or diamond drill bit can also be used to drill pilot holes for mounting screws, such as with mirrors or for towel bars. Place a piece of masking tape on the face of the tile to help hold the drill bit in place when you start drilling through the tile.