I think that if you're going to remove the whole case, some damage is inevitable. I used to go door to door in apartment buildings in the Jewish parts of towns, and you could always tell when a Jew had recently moved out, because there was still a mark on the doorpost (even when no screws or nails were used).
If you're worried about the way the doorpost looks while you're getting the scrolls checked, I have two suggestions.
Get mezuzah cases that allow you to slip out the scroll without removing the case from the doorpost.
Remove the case, take out the mezuzah, and put the case back up.
In both of these cases there is now an empty mezuzah case on the door post. I'm not sure if that's an issue. (If it is, maybe it's best to leave a note, or to have spare mezuzahs to put up whenever you have to check the regular ones.)
If you're worried about the condition of the doorpost after you move out, there's really no way to put up mezuzahs, that I know of, that won't leave some mark. The question is how much. Tape or other adhesives may require nothing more than some Goo Gone or the equivalent, while screws may require you to get some wood filler and then repaint.