Replacing a GCFI plug on my fan
If I understand correctly, you have a GFCI device on the end of a power cord attached to a fan - i.e., the GFCI was provided with the appliance and is not built in to the building. Assuming that is the case, there are two possibilities:


Failure of the GFCI
A ground fault in the device


If the GFCI has failed - and they can fail - then replacement makes sense. However, I think more likely is that there is a ground fault somewhere in the fan. A ground fault could include wiring (e.g., starting with cracks in the cord insulation), motor damage, switch damage, etc. I know of only two ways to get a 100% definite diagnosis:


Replace the GFCI. If the fan works, the problem was the GFCI. If the fan doesn't work, the problem is the fan - in which case toss the whole thing (and in which case you have just wasted a new GFCI module).
Replace the GFCI with a temporary regular plug. Plug the fan in to a GFCI-protected receptacle (e.g., kitchen or bathroom that has been updated to include GFCI) and see if it works. If the GFCI receptacle (or breaker) trips immediately then the problem is the fan - in which case toss the whole thing. If the GFCI does not trip and the fan works, then you should replace the temporary plug with a GFCI module.


I suspect an indoor/outdoor fan has a GFCI included because typical use may include outdoors, in the rain, plugged into a grandfathered non-GFCI-protected receptacle. So bypassing the GFCI protection, except for a very temporary testing procedure, is not a good idea.