When I set the heat cycle on my central heating/cooling thermostat in the auto mode the heat does not shut off after reaching the tempature set. The A/C works fine in the auto cycle. Do I have a faulty thermostat or could it be something else?

It could be that the thermostat wires for the heat are shorted.
It could be the thermostat.
Try disconnecting the R and W wires off the furnace low voltage wire strip and see if the furnace shuts off. If the burner shuts off the it must be in the wiring or stat.

Now if the blower is still running but the burner is off after the thermostat is satisfied then it is a different problem.