If this is a trailer and you don't have a generator running, the A/C thermostat set to "ON" wouldn't have much effect. There'd be no 120-VAC power to cause compressor or fan to run.
A Furnace, YES. It operates on 12VDC and would light if the LPG was turned on. If the LPG is off, then the fan would run off the coach battery till it died, unless you have a charging line.
LPG detector - Not sure about this one. Furnace attempting to light and not lighting would release a little LPG but most would go outside. Low trailer battery voltage could set it to beeping. Some will do that once past a certain age.