If you have a volt meter and are handy, you can determine what's happening. First, make sure all power is off to the furnace. For there to be a call for heat at the furnace, 25 volts AC is switched to the W terminal from the R terminal (Rh or Rc possibly) at the thermostat and then to the W terminal at the furnace control board via the wiring from the thermostat to the furnace. If the thermostat switch is working then your problem lies at the furnace control board. If the thermostat switch is not working, you may need a new thermostat.
Find where it is easiest to disconnect the wire to the W terminal... either at the furnace control board or thermostat. Then using your meter, measure continuity (resistance) at the thermostat between the R and W terminals as you move the temperature setting on the thermostat. On a call for heat, you should have continuity (zero resistance). When you move the setting to below the room temperature, you should lose continuity showing an open circuit (infinite resistance). Remember, all power off.