How do I replace old ceramic tile without pulling up old?

For best results it is best to remove the old flooring first. Anything you place on top will create an awkward transition and height difference, hence a tripping hazard to any adjoining areas. If there are any doors involved, you would have to cut the doors down too.

The height difference could also affect a good installation of your toilet.

I wouldn't suggest covering tile with tile. Removing the existing tile will be best, especially if there are any inconsistencies or dips, etc. Also, if this is on a floor, the floor height would then be higher and you not have door clearance or it may be higher than other flooring that it will butt up to. ... of if this is on a backsplash, it will overhang over your counters

If it's on the floor, place the new ones over the old. If it's on the walls, get some grout & fill in the lines to make them level with the original tiles to create a flat surface. Let it dry overnight. Glue a thin piece of wood over the entire wall. Start from the floor & glue your new tiles on. Grout, wipe off tiles with a damp sponge or cloth. Let it dry overnight. Done.

If you are doing tile over tile on a wall you do not need to grout and put a board over it. You paint old tile with Eco Prim. It creates a cement like finish on old tile and you add new tile directly onto it just as if the old tile was the wall board. Grout the new and you are good to go.

Are you trying to replace missing or broken tiles? Or cover them? If you are thinking of covering them several people mentioned level issues. Another problem might be the additional weight. You don't want to overload your floor. I'd try removing them rather then covering over them. I'm not sure why you would want to go over ceramic tile.

If you are trying to replace missing or broken tiles. Remove the surrounding grout. Clean the place where the tiles were very carefully and thoroughly. Try not to leave old glue or grout behind. The replacement tiles need to be glued in, you can use spots of mastic, once the glue is set you can regrout. There is a product that helps grout stick to porous material it might help new grout stick to old if you can find a perfect match. If it's on the floor you will want to reseal everything.