If I have the furnace on and the fan on "Auto" the fan never comes on. The furnace seems to work fine - the ignitor begins to glow, gas comes on, and burners light. After a while the gas stops for a time (before coming back on) but the furnace seems to run continuously

If I turn off my furnace and turn my fan to "On", the fan blows air fine. When I then flip my furnace to "On" as well, the furnace comes on fine (burners light), but after a few seconds the fan goes off and doesn't circulate any more air.

Any ideas what the problem might be?

The fan is controlled by a relay and a limit switch. When the gas comes on, the limit switch warms up, closes and sends power to the coil of the relay, which closes sending power to the fan motor. The fan on position sends power directly to the relay, but often a different higher speed relay starting the fan. Normally if there is a problem with limit switch or relay, the furnace will run with the fan in the on position. I am not sure why the fan stops soon after the gas comes on. These problems usually are with the relay, which are hard to diagnose and replace if they are on a circuit board. It this case, I am wondering if the problem is in the low speed winding of the fan motor. Try to check to see if there is power to the fan motor after the gas is on for a few minutes, but before it quits. Checking it will be much easier with a voltage detector.

I came across the niftiest gadget for trouble shooting, a voltage detector. They work through the insulation of wires. There are several brands. I have a GB Instruments GVD-505A, less than $15 at Home Depot. Touch it to a hot wire, and the end glows red. Find the doodad that lights it on one side, and not the other, and you have the culprit.

Thanks labman. I tried switching the wires from the fan, switching the black high speed wire form the COOL tab to the HEAT tab on the circuit board. The fan still does not work (even in the On position) when the furnace is on. I'm wondering if the circuit board is not sending power to the HEAT tab and there is some logic in the circuit board that says even when the fan in "On" power has to come the fan from the HEAT tab when the furnace is On.

I am not sure why they would have wired it that way if so. I would pull the low speed wire off the board and jumper power to it, and see if the fan runs. This will tell you whether the problem is the fan windings or the controls.