Ceiling fan on one circuit tripping GFCI on separate circuit
Try a Lamp Debuzzing Coil on the line-side of the dimmer

This sounds suspiciously like your dimmer is generating some sort of switching trash on the power line that's interfering with the GFCI's circuitry and causing it to trip, since it only happens if you change speeds.  The general fix for this sort of interference problem is to wire a Lamp Debuzzing Coil (LDC) (which is a prepackaged choke, basically) in on the line side of the dimmer, as documented in Lutron appnote #519.

If it persists, change the GFCI out for a new one of good quality

If the Lamp Debuzzing Coil doesn't fix it, then I would keep the coil in place (it'll cut down on RF trash coming out of the dimmer in any case) and replace the GFCI with a new one of good make and quality (you want a specification or industrial grade GFCI receptacle or deadfront device here, not the cheapest builder-grade thing you can find).