It was my CD player. An internal switch failure caused the CD player to go into a diagnostic mode. In diagnostic mode, the player stayed on even after I turned the vehicle off. To see if this is a problem, just pull the stereo fuse and see if the battery drain stops.

I was able to play with the buttons on the player to get it to stay off but then I had to be careful about not touching any of the buttons that activate the player. As soon as I pressed one of those buttons, the player went back into diagnotics.