I have a heat pump outside with indoor attic air handler. I don't use it very often. I tested it and found that the system seems to work when I turn the temperature up on the thermostat but when I turned it down again to shut the unit off, the heat pump turned off outside but the fan blower always runs unless I turn off the breaker. What could be wrong ? Indoor unit or outdoor ? Thanks !

When you turn tstat back down does the outside unit turn off? If so it has to be the inside unit. When you turn it down and it keeps running does it blow out cool air or warm air?
You might double check tstat and make sure accidentally the switch wasn`t moved from auto to fan or fan only
When it should be turned off and its blowing cool air you might have a stuck or bad fan control
Not knowing what kind of furnace/airhandler you have more info will be needed