Good suggestions, especially scoring around the old nut with a sharp blade to prevent chipping.
The most cautious way to remove a nut with no possibility of chipping fretboard or finish is to score all the way around the nut with an exacto knife, then cut almost all the way through the center of the nut lengthwise with a hacksaw or coping saw, pinch it with pliers until it breaks in two, and pull out the two pieces.
Never worry about destroying the old nut...there is a good reason that you are replacing it and you will NEVER reuse it.