Fey Ancestry just says magic can't put them to sleep. It doesn't cancel all magical effects that include putting them to sleep.

But it's more complicated than that. Imprisonment has the phrase "While affected by this spell, the creature doesn't need to breathe, eat, or drink, and it doesn't age." So, if you get rid of the only other effect (sleeping), does that mean they're not affected by the spell, and thus they do need to breathe, eat, etc.? Or does the spell affect them, because it still makes it so they don't need to eat, breathe, etc.?

Though you could argue that that's not the only effect of the spell. It also makes it so that you'll be detected by Detect Magic. Being an elf doesn't stop that, so you still won't need to breathe, eat, etc. Unless someone casts Nystul's Magic Aura.