Ok, without removing any wires.
Put the meter to DC volts and place the black lead to a clean ground. You can use the engine, or master cylinder or the front battery negative terminal.
With the red lead, touch the top post (big red wire), you should see 24 volts. (key on or off)
Next, touch the red lead to the small red wire. You should see 12v when the key is on(and I can't remember if it may have voltage with the key off. If it does that's ok)
Next, touch the red lead to the big orange wire, you should not see voltage unless the small blue wire is grounded. Ground it and see if you get 24 volts on the orange wire. It will suddenly drop to around 12 volts, thats what it does when the gps are on.
If you do all these tests, then the only other thing to see is if the GP controller card is supplying a ground to make the relay function. You can switch the meter to read ohms and Put the red lead on the blue wire(with it disconnected) to see if you get ground with the key on. If you don't, it may be your controller card or your temp sensor.
You can test the sensor for ohms(specification is in the diagnosing section of the TM(-20), IIRC.