I have an Ashe focused on the board and I remove a card from the opponent's hand with it. Ashe later dies, does its effect still trigger when the opponent holds or is the card banished forever?

Yes, the effect still triggers when the opponent holds, even if Ashe, Focused is no longer on the board.

According to the card text for Ashe, Focused:

"When you play me, choose an opponent. They reveal their hand. Choose a card revealed this way and banish it. When they hold, return it to their hand (even if I'm no longer on the board)."

Explanation:

• Persistence of the Effect: The card explicitly states that the return effect occurs "even if I'm no longer on the board." This is a delayed triggered ability created by the initial "When you play me" effect.
• Triggered Ability Rules: Once a triggered ability is created, it exists independently of the source that created it unless the card text specifies otherwise. In this case, the card text explicitly confirms the effect persists after the unit leaves play.
• Banishment: The card is not "banished forever." It is moved to the Banishment zone when the initial ability resolves, and then the delayed triggered ability moves it from Banishment back to the opponent's hand when the "hold" condition is met.

Note: I could not fully verify this ruling against the retrieved rules.