What would cause the rear brake lights to stay on constantly even when the car is off on a 1991 CRX?

Check your brake pedal, follow it up into the under dash. You should see where the pedal stop meets a bar with a switch attached. The rubber piece that triggers the swith to shut off the light when the brake is released probably broke off which leaves a hole in the pedal bar for the button on the switch to go through without being depressed to shut off the light.