I have a Frigidaire Furnace. I had the heat on for a few days and shut it off at the thermostat, but the heater kept on running. What could cause this and what needs to be replaced?

Either the relay or the capacitor on contactor is going/gone bad. Hard to tell which unless you test them with a multimeter tester and then change out the part.

The fan running for a while is normal because it stop only the heat producting part cool down.Then you check the setting of the fan,If you set it on it will continue run.If the heat continu come out.First try to complete turn the power off to reset the control unite,If still not work then the control panel need to be replace.