Furnace runs constantly and gets quite HOT, even if thermostat doesn't call for heat. Changed thermostat, didn't help. Removed new thermostat and checked to see if thermostat wiring had spark, it didn't. What now? I recently vacuumed unit. Does that matter? It ran fine for a few weeks following clean-out but not anymore.

You may have hit some low voltage wires and did not notice. Check to see if low voltage thermostat wires are touching in side the furnace R and W. You can also remove the thermostat wire from W on the furnace control board and see if it will run then.

I disconnected the thermostat wire from W on the furnace control board and it does NOT run. Now what?

Continuing the good start T-top gave you. Since the problem didn't seem to be in the thermostat, the Red and white wires must shorted together somewhere. Check the cable if you can, or it may be easier to pull a new one.