I have a 2001 XJR with an issue. The rear taillight stays on, regardless of whether the car is on or not. I took the bulb out to keep the battery from dying, but figured I would ask some amazing Jag enthusiasts before I waste my money on someone who paid better attention in their physics circuitry courses! Is it a short circuit? An open circuit? Do I simply need to replace a different bulb?
Any and all advice is greatly appreciated. Thanks!

Are you sure it is the "tail light" and not the brake light? Failure of the brake light driver in the SCLM (locking module) is common and you can find info in the archives.