In Keyforge, what happens to the Æmber on a creature that is archived?
All the Æmber currently on that creature will go to the opponent's pool.

From Keyforge's Rules

The glossary for Leaves Play reads (ellipses and emphasis mine):


  If a card that is in play leaves play (is returned to hand or deck, destroyed, discarded, archived, or purged), all non-Æmber tokens and status cards on the card are removed, [...].
  
  If a creature with Æmber on it leaves play, the Æmber is placed in the opponent’s Æmber pool. If a non-creature card with Æmber on it leaves play, the Æmber is returned to the general token pool.


Therefore, all the Æmber on a creature will be placed in your opponent's Æmber pool when it leaves play, not only when it's captured.