Choose your soap dish. Get a soap dish that matches your current tiles. Note how big the soap dish is relative to your tiles. If the soap dish takes up the same amount of room as one tile, it will be easy to install. If the dish takes up a fraction of a tile, you may have to cut your tile to allow for installation. The soap dish should have a protrusion that will fit in the recess on the wall where the old tile will be removed.
Score the old tile that is in the location where the new dish needs to be for removal. If the tile is to be completely removed, use a nail punch and hammer diagonally from top right to bottom left and then from top left to bottom right. Drill out holes in the tile that do not penetrate into the wall itself by setting the depth on your carbide-tipped drill bit to the same depth that you believe your tile will be.
Use a Dremel with a cutting disk to cut through the tile where you need it to conform to the shape of the soap dish and do not wish to remove the entire tile. Go carefully and take your time grinding the tile so that the tile does not split. Make parallel lines 1 cm. apart that cover the area that you wish to remove. Snap off the tile with a pair of pliers at the end of these parallel lines to remove small pieces until you get to the desired shape.
Scrape out any loose debris with a chisel and hammer behind the tile that you have removed to give the mortar a nice location to bond. Use a slightly damp rag to remove any dust that is still in the location. Scrub loose any flakes left behind without cutting yourself.
Apply a generous amount of an adhesive mortar (like Thinset) on the back of the soap dish. Press the soap dish into place so that the dish presses the mortar evenly into the crevice. Leave a gap around the dish for the grout. Use a level and plastic tile spacers to set the soap dish evenly. Tape the soap dish to this spot with a generous supply of painter's tape. Clean off any excess mortar. Wait for the mortar to cure.
Remove the plastic spacers. Add grout around the soap dish using a popsicle stick. Scrape the grout to the desired level so that it matches the level of the grout around it. Use a damp rag to wipe up excess grout. Allow the grout to harden. Apply a penetrating sealant to the grout, and allow it to dry to prevent water from penetrating the grout.