Stuck relay or g tstat wire is touching hot 24 volts somewhere.
Depends what kind of air handler or furnace you have. generally it will be a fan relay stuck.
SOURCE: thermostat won't keep furnace fan running
You probably have a loose or broken wire to the fan relay. Check the wires on the thermostat first. Pull up on the bottom of the thermostat to remove it from the wall plate. You should see several screw terminals and the conductor wires that should be connected to them when you look at the wall plate. Check if any of the wires are disconnected or broken. If a wire is broken, strip a portion of the unbroken section of the wire and secure it to the appropriate screw. (The fan relay terminal is G for most setups.) If none of the wires are loose or broken, then you have to check for damage to the wires between your heating system and the thermostat or at the heating system. It could be a bad wire going to the jumper on control board or damage to the board itself.
I hope this helps. If the problem is not at the thermostat, you may need to have a professional diagnose and fix the problem. However, if you post additional comments about your heating system, Fixya experts may be able to help you diagnose the problem.
Cindy Wells
(we had the broken wires at the thermostat keeping the fan from working in the past. It's an easy fix if you are willing to handle small wires carefully.)
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When this happens, the first thing I was taught to do is to pull off the tstat face, and read for voltage 24vac across common and the other terminals. If you read 24+/-3v, you have wires touching, excluding R&C (that is how the tstat energizes the other terminals), bleeding over. Any service, or install tech will perform this test with a similarity to that symptom, regardless of line of work they are in.
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