I just went through a bunch of trouble shooting on a water heater in my RV. I'm a very good low voltage tech and worked tech support for many years for a electronic security mfg. However my experience with RV's are limited but I do suggest the following:
1) Connect one lead of your volt meter to a good known neg or ground source and set it for DC range around 12vdc
2) Since we know the sequent of the start up for a furnace, follow that with all the wires connected
3) Turn thermostat to heat and put the other lead of your meter to the motor pos+ lead and see if voltage is present. If so then why isn't the motorr moving ? Search that. If relay clicks, then what is feeding the relay common side ? See why there is no voltage coming across the relay contacts ?
4) If the motor starts then check both sides of the sail switches to see if they are working correctly. If OK then why isn't burner starting.
5) Ck pos+ leads to gas valve and see if there is voltage there, etc
* Just follow the logic trail and see where it fails is my suggestion ...