My gas furnace is working fine, but the small fan is kept running all the time. Some one told me that the circuit board causing this problem. Is that true? Thanks

Post a brand and model number...

I believed it was Carrier 8000 the name tag is missing.

Is the inducer motor running even when the thermostat is in the off position.

That's correct.

You must have a relay stuck for it to run when the thermostat is in the off position, I will do some reading and see what I can find as to why this would happen.