The blower on my heating unit will not turn off, either when the thermostat is set on auto and the temp is exceeded, or when the unit is turned off at the thermostat.

Depending upon the type of furnace you have it will be a relay,sequencer,or other device that has failed.

OR

The unit has become overheated and triped the high limit safety control. In that case it will have to be reset or replaced. There is a wiring diagram located on your unit since they all have one. Usually they are located on the back side of the blower motor access door. Folow it to find the parts in your furnace that could be suspect.

Make sure your filter is clean and all registers are open.