Detaching large ceramic tile without breaking it
I have had (some) success using the following technique:


using a utility knife or grout saw, remove as much grout as you can
using a narrow margin trowel (see pic) with a hammer, carefully tap the trowel under the tile in as many places as you can. This breaks up the mortar somewhat. Do not use any leveraging type action until you have tapped the trowel under the tile around as much of the perimeter as possible
carefully start using leverage with a wide margin trowel in various spots around the tile until it (hopefully) pops free in one piece