My Central Heating and Air unit just started having this problem. The fan will not turn off unless I cut the power completely off manually. Everything works normal by the thermostat except the fan. When I have the thermostat set on auto fan, the fan speed slows down and when I switch it to on, it runs at full speed. This is being determined by the sound. When I disconnect the thermostat the fan still continues to run but only at full speed. Could this be that the relay inside the metal box with the transformer that's located on the blower fan unit need to be replaced? Any help will be appreciated.

It could be a relay or a fan control.

Is it gas heat or electric? If its electric it may be a fan sequencer or relay stuck, if its gas it maybe a fan limit control switch stuck. either or it will not hurt the system if the blower runs all the time. give more info if you can.