Sounds like you have a 'short' in the wiring somewhere. Broken or chafed wires most likely. I would start from the bike of the rear light unit and working forwards, check all the wiring relating to the light unit.
I'm with thelargeone on this. It sounds like a dead short to earth. Time to get the old electric string chart and tester out and start checking through the multi-coloured spaghetti.