The blower final stage fan resistor is known to fail. Symptoms are the heater fan speed varies regardless of the setting, the fan will not turn off when the car is switched off, or the fan will not switch on at all.Google+
Car Problem Reports
BMW 540i Blower Final Stage Fan Resistor May Fail
BMW 540i Problem
Average Mileage: 102,572 mi (10,009 mi - 202,000 mi)
Visitor, 2001 BMW 540i, 153,000 mi
Blower motor continues to run even with the ignition off, and key fully removed. (it's really freaky).
Found parts online $60, readily available. (My local dealer, BMW Concord, CA. quoted me $193 for the part), found video on YouTube how to replace, took about 20 minutes.
Visitor, 1998 BMW 540i, 105,000 mi
Blower resistor went out, fan control eratic. Replaced for $70
Dealer replaced fan transistor (I think) due to intermittent failure.
Visitor, 2003 BMW 540i, 39,000 mi
Fan speed allways on low.
Caused battery to go dead. Replacing part fixed problem