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.
Car Problem Reports
BMW 330xi Blower Final Stage Fan Resistor May Fail
BMW 330xi Problem
Average Mileage: 133,414 mi (115,000 mi - 155,244 mi)
Visitor, 2003 BMW 330xi, 155,244 mi
IT WOULD TURN OF WHEN IT WANTED THEN NOTHING THEN ON NOW NOTHING
Visitor, 2001 BMW 330xi, 130,000 mi
Every time I turn off my car there is a faint sound of a fan running inside the car. I would assume it is the blower fan for the heating and cooling inside the cab. I haven't fixed it yet but replacing the final stage fan resistor seems to be the common fix. I'll try that first.
Visitor, 2001 BMW 330xi, 115,000 mi
Final Stage Fan Resistor replaced