Originally Posted by labman
I copied this from an answer I gave yesterday to a similar problem.
You do have a furnace, not a heat pump? It is fairly common for the blower relay to stick, but it would a real coincidence for the heat valve to stick too. I am guess it is the thermostat, cheap and simple to replace. First, try switching it to cool. If that doesn't work, shut the power off and remove it from the wall. Remove the red and white wires noting if they come off terminals marked R and W. With them not touching, turn the power back on. If the heat doesn't come on, the problem is in the thermostat. Note, the burner will come on until the furnace heats up and the the blower, which also runs a little after the burner goes off. If the heat comes on, go to the furnace and disconnect the white or red wire. If that fixes the problem, it is the wiring between the furnace and thermostat. If that still doesn't stop the heat from coming on, likely the control board is bad. Harder and more expensive, but still can be a DIY project.
Check the things I suggested, and post back if you need more help correcting the problem once you identified it.