If you play a duration card that has been turned into a pig, does it stay in play the next turn?
I see the game isn't doing that.
It does not stay out because you don't resolve its usual effects.
That's correct.
Durations cards stay in play until the turn they 'do something' (Wharf is lets you draw 2 cards the next turn, Archive is giving you the last card stored under it, etc.). If a card is 'enchanted', the only thing it does on playing is '+1 Action / +1 Card'. So there is no effect set up to do somthing later, so the card is discarded during the turn's cleanup, because it isn't going to do anything in a later turn.
The same is true in rare situations whithout Enchantress. If you play an Archive, but have only one card in your deck/discard pile, you will get that card into your hand, but Archive will not stay out, because there are no cards stored under it, so no cards can be 'given out' in later turns.