Diagnosis
A ground fault or serious current leak inside the microwave. The usual suspect is moisture ingress or a component failure in the high-voltage system, like the magnetron or capacitor.
How to Fix It
First, test the microwave on a known-good, non-GFCI outlet on a different circuit. If it works, your kitchen GFCI may be overly sensitive or failing. If it still trips a different GFCI, the fault is in the microwave.
If That Didn't Work
The internal fault requires disassembly. Check for obvious signs of water or rodent damage near the power cord entry and control board. The magnetron or high-voltage transformer can develop internal shorts to ground.
Heads Up
Do not bypass the GFCI outlet. This is a critical safety device detecting a potentially dangerous fault. Continued use could lead to shock hazard or fire. This often requires professional repair.