[Evgeni]

It would most likely be a high ground on your tail light circuit. When the lights are off, the light is drawing from a shared ground in the circuit. Turn the lights on and there is no ground available probably due to a partial short to power.
You need to test the power/ground at the bulb socket.
Solutions for should it be a faulty ground would simply be grafting a new body ground to the bulb socket or to replace the tail light wiring loom from a wrecker, provided the bulb socket is all ok.