Note that, as others are pointing out, the screw is captive, so it will not come all the way out--when you loosen it the door will come off, but the screw stays with the door. But it sounds like you just can't get the screw to move.
...which is good, because it means you're not stripping it. Which also implies that the screwdriver is at least close to the right size for the screw--if it's wrong, you run the risk of stripping it, but the screwdriver would be popping out rather than just not moving. Therefore, like rpaloalto says, bigger handle screwdriver so you can turn harder. And make sure you push in hard, so it doesn't slip out and strip it.
Hopefully you won't have to resort to ice or WD40 (both of which might help). There's also Lube Job or any of a variety of thread looseners you can buy at your local hardware store, but those are nasty stuff unless you're desperate.
Related tip if you do have problems with the screwdriver coming out of a partially-stripped and very tight screw, taught to me by a Xerox maintenance guy: Take a correctly-sized screwdriver bit that's nice and sharp, put it in, and tap the screwdriver in with a hammer. This embeds the bit in the screw a bit and gives it something to push against. Obviously that should be a LAST resort when it comes to something as fragile as an expensive computer.
