Why does my central air/heat blower fan come on when the thermostat is turned off. I even took the thermostat off the wall (it is a programmable Honeywell thermostat) and it still comes on at random times. May run for 1-5 minutes then turns off.

Could be the relay (fancy word for switch) in the air handler or area of blower motor. If the contacts are closing without any signal from the thermostat it could be your culprit.

Also check for wires, specifically green and red from the t-stat that may have a bare spot and are rubbing together although I put the odds on this lower.