What is the easiest way to remove a laminate backsplash and do the least amount of damage to the drywall?

Been there and done that.
If it is glued or secured directly to the wall, I would just cut the drywall around it and remove it attached. Trying to pry it off or save yourself drywall repairs seems to never work. (at least in my limited experience)
(Might be a good time to check for any insulation issues or needs if its an outside wall too.)

Plus if you are planning on tiling the back splash area, it can be a great time to run wiring for under cabinet lighting with switch or extra outlets depending on whether it is a kitchen or bath.

also had the experience. one little pry test with a putty knife, and I decided we were going to replace all the backsplash area drywall.

one, we went for a tougher tile backer product. two, we added some outlets and replaced the boxes and doubled up on all of the countertop outlets, two devices instead of one. we also ran a LV power wire to a hidden-by-microwave outlet for an alarm system, and jumpered over telephone wire for the system.

it also allwed checking up on the state of the insulation and vapor barrier on the exterior wall.

really, if you've got that situation, a cut and replace is the way to go.