The fan control on my Carrier Thermidistat does not turn the blower fan on when the Fan button is set to On. In Auto, the fan will not operate the fan on either a heating or cooling cycle with my heat pump. The fan DOES work when the furnace operates in emergency mode which is being controlled by the thermostat on the furnace itself. It just seems that fan operation through the Thermidistat is broken. Is it time for a new tstat?

NO you probably have a bad relay.