Bill, I believe the ignition controller box is activating the fan, it clicks when the fan comes on. Are you saying a bad limit switch at room temperature will be open or closed?
At the root of the issue is this: the fan motor won't come on with the burner operating. The relay clicks, the gas fluctuates, but the fan does not come on. I'll try to break it down to make it more easily understood.
1. Gas burner ignites and heats
2. Relay clicks in ignition controller, gas fluctuates, fan DOES NOT operate (happens approximately 3 times)
3. Gas burner shuts off
4. After a brief pause, fan activates
5. Fan runs for a short period of time, then gas burner lights
6. Heater runs for a while, shuts off, then does not come back on
That's another point, after it runs through one cycle, it does not come back on. I can normally tell it's not running because the house starts getting cold.
Would the heater not coming back on also tend to implicate the ignition controller?