Why does the HVAC thermostat say that the heat is on when it isn't and the air conditioning works perfectly?

All that thermostat indication is saying is that the heat is supposed to be on. If you hear the fan running for a minute or so after the heat indicator has gone out this is normal as the fan is blowing out the remaining residual heat in the furnace after the flame has gone out. The same works in reverse once the stat indicates a call for heat the fan does not start for a minute or so to allow the furnace to heat up and not to blow cold air into the space at start up.