I have a heat pump system and a programmable thermostat. The system runs fine, in heat or cool mode, with the fan set to 'auto'. It will not run (no fan) in heat or cool mode with the fan setting set to 'on'.

Most likely the fan only wire is not connected or connected properly. Has this always been a problem or is it a new problem.

Just curious, why would you want the fan to run continuous.

Some people are like that,, I wonder why also,
Some seem to think that recycling the indoor air saves money

I can see some reasons for it, but I have found the most profound outcome is shortened life of motor.