Can a GFCI burn out a (ceiling fan) motor?
From the perspective of a motor, a GFCI looks exactly the same as any other circuit breaker.  The only difference is that a GFCI has extra circuitry which senses an imbalance between the black and white wires (plus control circuitry to trip the breaker)&mdash;nothing more.

Many modern ceiling fans seem to be cheaply made.  I suspect the failure is a quality control issue or an intentional cost cutting measure of the manufacturer.  If it is still in warranty, get it replaced.