So, this indicates to me that I was running my furnace without a ground, which sort
of (sort of) goes against claim #1 that the control boards need a good ground to
operate properly. Of course, maybe my furnace *seemed* to have been operating
properly with the generator but really wasn't, or maybe my control board doesn't
even need a ground.

Comments?

Excellent question...

the usual reason the controllers need a ground is for flame sensor. This usually applies to GAS fired devices. The electronics senses the flame as it touches metal parts that are grounded. If the electronics does not have a good ground, this may not work reliably.

The solution would be to use a three prong plug with a ground to connect the furnace to the generator. Connect the white and black wires as you did. Connect the green wire to the ground in the outlet box where you tapped in which should also connect to the frame of the furnace.

If the flame sensor does not work correctly, then add a jumper wire to connect the white and green together.

My generator actually already has the white and green tied together at the outlet box.

If your furnace worked OK as it was, then I wouldn't worry about it. THe flame sensors need only a high impedance ground to work correctly. If it ain't broke, don't fix it.

Mark