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Dungeon Geists wouldn't even be able to target the creature with hexproof, so you would simply have to pick another target, or if no other target exists, then you don't get any part of the ability.

To answer your question directly, Dungeon Geists would not be able to keep a hexproof creature that is already tapped, tapped down for as long as Dungeon Geists remains in play. Again, you would need to pick a different target an opponent controls (as the ability is not a may), unless there is no other legal target, in which case the ability fizzles as you say.