Hammer it out.

use protective goggles.

When my wife and I decided to have a new granite counter in our kitchen my problem was
somewhat different with the backsplash. I wanted to be able to temporarily remove the first
(bottom) course of tile to save and put it back after I had the new granite in place. I accomplished
this by using my wife's hair dryer and warming up each tile before I popped them off using
several thin flat bars. Worked like a charm and left me with minimal drywall repair. This method
will work well as long as your tile hasn't been installed using thinset. If thinset was used, you
might as well plan on replacing the existing backsplash drywall

Peter