If the resistor burns out, it was not sized correctly to begin with....(wattage wise) Basic electronics 101.
The resistor is there as a cheap and dirty way to lower the speed of the fan. The current flowing through the resistor creates a Voltage drop across the resistor. I measured 14.75vdc across the battery terminals....(eng idling) With the resistor in the circuit, the measured Vdc across the fan is a lot less. A relay shunts out the resistor for full voltage / max speed ) aka high speed).
If the fan won't handle continuous high speed operation, it too is not designed correctly.