You most likely know this, but after three attempts to light the gas, the furnace will "lock out" to prevent gas from building up in the furnace and causing an explosion or fire if it finally did light.
There are many things that can cause a "no light" condition; but just like any appliance that uses fire, power, fuel, air, the right mixture, and an ignition source is required.
12 volt Power - To get the furnace to run, good strong 12 volt power is required. This means a known good fuse, good 12 volt wiring from the fuse block to the main 12 volt connection on the control board, a good thermostat calling for heat and solid wiring from the thermostat to the furnace control board.
Fuel. To get propane to the orifice:
1) There has to be enough propane pressure to feed the biggest consumer of propane in the camper. If you have all the burners on the stove running on HIGH with tall blue flames; this is good.
2) There has to be enough 12 volt DC power to run the system. A low battery will cause several problems we will also talk about later when we talk about "air." The gas solenoid need a fair amount of DC power to open and stay open. The DC blower motor needs a lot of DC power to run. The control board needs a minimum of 11.5 volts as well. The furnace fuse has to be good.
3) To get gas to the orifice, the motor has to turn fast enough to suck air from the camper, into the squirrel cage fan to force air through it and back to the camper. There is a small tap of plastic attached to a microswitch that has to "see" enough air pressure to close, completing a circuit to the control board. If this switch is bad, there is not enough air flow from the fan due to dirt or bugs nest, the fan motor is gummed up and turning slowly, or the battery voltage is too low to spin it fast enough; the "sail" switch will not close and the furnace will shut down. There is a timer circuit in the control board that "waits" for that signal for about 1-2 minutes before it shuts down.
3) Assuming the control board "gets the OK" from the sail switch; it sends the "open" voltage to the gas solenoid. The gas solenoid will then open a release propane into the combustion area through a carefully sized jet (orifice) for the altitude (normally sea level). Adjustments to the mixture for higher (or lower) altitudes are made with the air shutter to adjust the mixture.
1) Combustion air is supplied by the SAME motor that supplies the interior air flow. The motor shaft has a squirrel cage fan on each end of the motor. If the motor is not turning fast enough for the blower; it is not moving enough air to light the fire either (over rich mixture).
2) The combustion exhaust has to exit the camper through the flue. Other wise the exhaust gasses will stay in the combustion area and put the fire out.
The lack of oxygen in the combustion area (rich mixture) will cause a 3 light attempt lock out. Bug nests (mud daubers/wasps) and dirt are the typical suspects here.
Mixture: Must be exactly correct for proper operation.
1) see above - People sometimes try to fix dirty orifice jets with a paper clip (don't do it). The steel will damage the soft brass and the overly large hole will cause a "rich mixture." Dirt in the hole will cause a "lean mixture." Lack of proper exhaust will cause a "lean" mixture. All of these can cause a "no light" lock out.
1) The control board - The "brains" of your heater is the control board. It takes the low voltage "go" signal from the thermostat; routes much higher amperage power through a relay to the blower to start it spinning; waits for the sail switch to close; routes power to the gas solenoid and ignitor; and monitors the exhaust temperature for a rise. If it does not see a rise in temp in the exhaust (using a bellows type temp switch), it shuts down the start attempt by closing the gas solenoid, purges the gas from the combustion area by keeping the blower motor running, reopens the gas solenoid and starts the ignitor again. After 3 cycles with no temp rise you are done until power is removed from the furnace by pulling the fuse at the fuse panel.
2) Obviously the ignitor itself can be bad. There is a spark tip just like in your BBQ that needs the proper gap to make a hot enough spark to light the gas.
This is a LOT of troubleshooting. Worse yet you will need to remove the unit from the camper to get to most of these parts. There is an upper limit as to how much a technician can charge you to fix your furnace. If the repair (which includes the cost of travel to your site, troubleshooting - as best you can without replacing anything, the bad part - or parts if you guess wrong), is going to cost more than half of a new furnace you will not be a happy customer.
I think this is why no one wants to travel to your site to work on your heater.
Suburban RV Furnace - NT Series on Sale - PPL Motor Homes
Wildcat fifth wheel rv heater removal - FAQ Copy
has instructions for getting the heater out if you know a competent mechanic willing to "give it a go"