What causes the blower motor on my furnace stay running?

Limitswitch/cam stat can do that. Also t-stat with the fan switch turned to on.

If you can give us a make and model it helps us know how the blower is controlled. If the blower is on all the time but it is heating and controlling the temp. I would say the blower control is stuck closed or you have the fan switch on the thermostat in the on position and not the auto. If not keeping the temp in the house and the blower runs all the time it may be a open limit.-----Furnace or Heatpump?