Do you have to move toilet in bathroom?

to put down stick down tiles? Or do you just cut around the shape of the base of the commode and caulk after installation?

If you do NOT remove the toilet, then keep some tiles because you will need them to fill the hole if you ever have to change the toilet. You will NEVER find a new toilet with the same shape as your old one.

It makes for a much neater job to pull the toilet. WHen replacing it, you'll need at least a new wax ring, and it's not a bad idea to replace the supply hose (if it's a solid riser, it'll probably be okay, but since you've raised the toilet the thickness of the tile, you may have to either shorten it or (probably) bend it a little). You may need to replace the toilet bolts that hold it down if they were steel and have rusted since they may not come off. Use SS or brass new ones.

Other than being heavy, resetting a toilet isn't really all that hard.

It would also be a good time to evaluate your flange. The original flange will set too low with the added tile and will require adding a thicker wax ring or flange extender rings. If the old flange has signs of breakage or otherwise doesn't appear like it will last the 30 plus years, it would be a good time to consider replacing it so the flange sets on top of the tile rather than below.