I have a Hamilton Bay Ceiling Fan that is probably 12 years old. The motor is burnt out. I don't have the receipt anymore. Don't they have lifetime warranties? If so how do I go about making a claim with only the burnt fan?

I cannot speak to the warranty, but given the age of the fan and the cost for a new fan, I'd just replace the fan.

Don't they have lifetime warranties? If so how do I go about making a claim with only the burnt fan?
This is really a question for them, wouldn't you think?
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If I recall, this is a Home Depot product. Bring the dead unit to the store nearest you and raise a stink. It may get you somewhere; it may not.