In Skyrim, do potions like Resist Fire have any effects on your followers? If I drink a Resist Fire potion, does that also increase the fire resistance of my followers?
No. Imbibing a potion or casting a beneficial spell on yourself has no property of transitivity to your follower. (Follower is the official term, if only because "Companion" is used to refer to... well, one of the Companions)
I'm not sure under what situations NPCs drink potions in their inventory, but it does happen (or at least is 'faked' from time to time).