Some fans are triggered by a humidistat, if this is on one, it's possible the high humidity has closed the circuit. But a humidistat in a powder room isn't the most logical location for an installation. You could try localizing the circuit that it's powered from by turning off the breakers in the main service panel one-by-one. IF the fan turns off when power is removed from that bathroom, if you're comfortable, you could also remove the bathroom switch face plate and see if it's simply hot wired, or shorted. This is a basic bathroom ventilation fan and not an air exchange fan or a heat recovery ventilator, etc? Again, a powder room would not be a logical location for ether of those. If you;re not certain, you could pull the grill off the fan and see if there is a manufacturer and model #. Then there is also the phone call to the previous owner asking what the deal is with the fan.

C&R - I believe you live in Texas? Well, Chicago doesn't have the same climate as Houston (thank god). Exhaust fans in the laundry are not required in my burb. And I have to say, I've never noticed that the washing machine raises the humidity measurably. My washer is a front loading HE washer and uses very little water. Never have I ever felt the need for an exhaust in my current laundry. Plus, my new laundry has an operable window. Anyway, it seems to me an odd requirement - don't most dryers essentially act as exhaust fans when they run because they take air from the room and exhaust it (through the dryer) outside? kudzu9 - my husband agrees with you that a 5 foot long duct does not impact a fan negatively. But most of the time this fan is not running, and during extremely cold weather the duct's air temperature (being essentially open to outside) drops to be freezing cold, and of course the longer the duct, the further that cold air reaches underneath our master bath floor, which is why he'd like to keep the duct as short as possible: we like warm toes :-)

Not an expert but sounds as if while the burner is on the flue gases are going up the chimney. When the burner cuts out the chimney effect quits and the blower is producing a negative pressure on the chimney or a leaking heat exchanger pulling the exhaust gases into the system. Again no expert but a leak is indicated. I think the CO detectors will not detect until a minimum threshold. The limited amount of time to possible CO exposure after the burner cuts out means the detectors may have insufficient CO present to set off alarm.

I would check with an electrical supply house. I would be amazed if a replacement was not available as this should be a still common type and size fan.