Sounds like there is a short in the radio.. If you undo the mounting screws for the radio and pull it out there might an obvious wiring issue (bare, cut, etc) that you might be able to fix yourself.. If it all looks good, you could undo the + wire from the radio and then replace the fuse.. If it holds the problem is in your unit and you may need a new radio..
Have you got a test meter? Do you have power at the furnace?
Camped 2016 (Days/Nights) - 32/23
Nights camped 2015 - 29
TT: 2013 26' Revere TBQB
TV: 2014 GMC Sierra 1500 Z71 4x4