Getting associated with a low prestige topic can lower someone's status even if they are on the side that isn't stupid. (See OvercomingBias.)

If picking up a hot iron can be considered to be advocating a stigmatised contrarian position then even just mentioning the topic without advocating for it could perhaps be considered 'touching the hot iron rather than picking it up'. (I think Eliezer made this up on the spot by expanding on the metaphor that you provided.)