That wire feeds the ignition coil and ignition module in the run position. During cranking, the wire is bypassed with another wire (see your diagram). When you let off of the key, the bypass goes away so the engine stalls.
The resistance wire is used to drop voltage to the coil and module. Full 12 volts will eventually damage both. During cranking, the voltage on the bypass wire is below 12 (due to the load of cranking) so no damage is done. So yes, you need the resistance wire, but I don't know if RS resistor will get the job done.
Bigger question is why is the wire getting hot now? Are you sure you hooked everything up the correct way?
I start what I finish!