GPS satellites are placed pretty high up, around 20,000 km — high enough that it'll take a very long time for their orbits to decay to the point that they're falling out of the sky. It's more likely that the satellite will just stop working first, and even more likely that it'll lose contact with the ground tracking stations before that.
The Global Positioning System essentially works by triangulation. For that to be reliable, each satellite has to know where it is in relation to the Earth's surface and at least three other satellites (they're arranged so that there's always six satellites in view from any given point on the planet's surface). They have to be synchronized from time to time to update their tracking information, since they do shift around in orbit a little (Long story short: No orbit stays exactly the same for very long). I'm not sure how automatic this process is, but if the ground tracking stations go without maintenance for a while, it's likely the computers are going to break down eventually. You might still be able to get a GPS signal with the right receiver if you have power, but if the tracking information hasn't been synchronized for a long time, it's probably not going to be very reliable.
edited 17th Feb '13 11:47:10 AM by Specialist290