Problem removing towel rack base?

There is an old towel rack (3pcs) attached to the ceramic tile in my bathroom. The part attached to the tile has a very tiny screw on the side of each holder. I can't get enough leverage with a tiny screwdriver to losen it, because it is so close to the wall. The metal part is very lose and you can turn it 360 degrees so the rod always falls out, so this holder is useless. I don't know if these tiny screws are what is actually holding it to the wall or if it is something actually on the back of the device that goes into the wall. I do not want to damage the tiles if at all possible. Does anyone have experience with this type of towel rack and how can I remove it. Can some HELP. Thanks