I haven't done the resistor mod but it is probably set up like everything else...
The ECM supplies a 5 volt reference voltage down that wire. It is measured in milliamps. It is supposed to be grounded to a certain degree, and that is supplied by the resistors in the shifter switch. The resistor actually lets some of the current through to ground and the ECM "senses" that. The less resistance the more current is grounded and you have a voltage drop in that wire, in other words it would go to something less than 5 volts.
So... in this example the ECM is designed to be grounded through that wire. If you were to ground that wire completely it would not burn up the ECM... it doesn't care if that wire is grounded. What would happen is that the ECM would get an incorrect voltage reading in that wire and it would think that you are in a gear that you probably are not in. So your bike would not run at it's absolute best, but chances are you would notice very little difference.
I'm not even sure if that wire were to short to ground if it would throw a check engine light. The Triumph ECM may not be that sophisticated.
As the Piggr said, if the resistor burnt up you would more likely than not have an open circuit in that wire, not a short. So nothing would happen. But once again, the ECM may think your bike is in a different gear than it really is.
I think in the greater scheme of things this is a pretty safe mod to do. You could probably do more damage by hitting a squirrel or hooking up the jumper cables backwards