Installed a ceiling fan dimmer switch in a bedroom that is on an arc fault circuit. When pushing the dimmer switch up or down on the ceiling fan, it then trips the GFCI outlet on the bathroom circuit. These are two separate circuits with two separate breakers in the breaker box so I'm perplexed as to why a GFCI outlet on one circuit would trip as a result of something on another circuit. Should I assume something in the breaker box is wired up incorrectly?
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).
I had the exact same problem - ceiling fan would trip two GFCIs on a totally different circuits. I was going to test all the components all the way up to the ceiling fan to fix the problem. Replaced the 3 position fan speed control-Problem persisted. Replaced the GFCI with a different brand of GFCI - PROBLEM FIXED. The original GFCIs must have been way too hair-trigger sensitive and tripped maybe due to the fan motor load. The new GFCI brand is "Legrand".