I can't see why it would need to be on a GFI if the metal parts of the
fixture are properly connected to the supply ground lead.
I'd expect a good chance of nuisance tripping a GFCI from condensed
moisture if the fixture wasn't perfectly sealed.
When our home was built the electricians powered the Nutone bathroom
ventfans/lights from the output side of the GFCI outlets installed in
Those GFCI outlets triped after almost every shower steamed up the
bathroom. I moved the feeds for the fan's timed switches to the input
side of those GFCIs and they've been working just fine for the last 20
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