[Nimrod]

First thing to try is replace the thermocouple if it has one. If it's dying it acts like what you describe. Also this is the simplest and cheapest thing it might be.

 

[Nimrod]

Another thing I have experienced is that this happens when the condensate drain freezes up.

The pressure of the propane in the tank can be very high depending on the temperature. One hundred fifty PSI is not unusual. The regulator on the tank is a high pressure regulator. It cuts the pressure down to somewhere between 10 and 60 PSI. You need this pressure to push the propane through the line to the house. Here, in my county, the tank needs to be at least 10 feet from the house, most are further. The regulator on the house cuts the pressure back to a little more than 1/3 PSI which is what the appliances run on.

You have tested that the stove works fine so the line to the stove is good. The furnace burns more propane faster than the stove so it may not be getting enough. To be certain you will have to hook up a pressure gauge. The pressure at the furnace should be a little more than 1/3 PSI AKA 11 inches of water.