How do you test a 240 line voltage electric baseboard heater thermostat.

With extreme caution. If you could safely get to the 4 screws on the thermostat, incoming would be 240 between 2 screws, and in order to see if stat works you would have to be in a cold room with the stat on and measure voltage between the other 2 screws. A bit dangerous for the inexperienced though maybe.