A thermostat fan that won't stop running regardless of how it is set is a common problem. Whether you've wired the thermostat yourself or a professional completed the job, fan problems tend to arise more than anything else with thermostats. Diagnose your situation accurately to alleviate the issue; otherwise, call a professional for help.
Cut the power to your thermostat through the circuit breaker. Turn the lever for this portion of the house to "Off."
Remove the thermostat from the wall. Pull off the cover and unscrew the back plate.
Leave the wiring connected the way you have it to check its accuracy. If you've wired the system yourself, inaccurate writing is most likely the culprit.
Check each terminal and wire to ensure they've been set up correctly. The green wire is typically your fan relay and should be hooked to the G terminal of the thermostat. The yellow is cooling (Y terminal), the white wire is heating (W terminal) and the red wire is the power source, which is generally the R or RH terminal.
Disconnect each wire by unscrewing the terminals and reconnecting it to the proper terminal. If you've noticed a mistake in the wiring, reconnecting the wires should correct your fan problem.
Screw the back plate of the thermostat back on and return the cover. Flip the power switch from your circuit breaker to see if the thermostat and fan start working correctly. Ensure the switch is set to "Auto."
Check your furnace blower if this is a winter/heating issue. Remove the panel from your furnace and look for the silver box called the fan time limit. There should be a white switch/button sticking out of the plate. Ensure this switch is pulled out and set on "Automatic." Consult a heating and cooling professional if this doesn't alleviate the problem.
Consult a trained professional for diagnosis if the thermostat wiring/settings and the furnace time limit aren't the problems. Don't attempt to change anything else, or else serious injury or damage could result.