When you say it shuts down after 5 minutes does the burner and fan shut down? Or does the burner shut down but the fan continues to run until you turn the thermostat off?
Are you plugged into shore power when this happens, or are you running off the battery alone?
Is there black soot on the outside of the trailer around or above the furnace exhaust port?
Can you find and tell us the model number of the furnace?
Do you have a digital multimeter to be able to take some readings?
Does your thermostat control only the furnace? Or does the thermostat control the furnace and the AC?