Actually, those screws are there for a purpose, but without them you wont cause any problems unless some one sits on the door when its down.
To fix this properly, I would not mess with the holes any more as they will only get larger. This stuff is glue and sawdust. I would pull out the dishwasher, not unhook any thing, just pull it out enough to be able to do this procedure:
Get a 1/4 inch piece of ply wood about 18 inches long by 4 or 5 inches wide. Apply liberally liquid nails to this board, but first start some 5/8 inch drywall screws in the plywood, then screw it in and let it sit for a night. then screw the dishwasher screws into this ply and you will be set. You should have ample room between this new board and the top of the washer. The 5/8 inch screws are short enough not to protrude up in the surface of the counter.