Sounds like there was a short somewhere on the board. Logic circuitry/the microcontroller on a board will work on 5 or 3 volts, powered by a voltage regulator. Wired sensors run on 12 volts, again sourced by a voltage regulator which would probably source 1 or 2 amps.
The resistor here seems to be in the power /supply battery section. If the resistor has burnt out, chances are some of the other electronics became faulty and caused this to happen, so even if you replace it, the it may blow again.