If I gust an Irresistible Faefolk with its trigger on the chain, does my unit still get moved?

Yes, your opponent's unit will still be moved to the battlefield.

According to RiftJudge FAQ #10923, when Irresistible Faefolk’s ability ("When I move to a battlefield...") is placed on the chain, it locks in the reference to the battlefield where it arrived. If the Irresistible Faefolk is returned to its owner's hand by Gust before the trigger resolves, the ability remains on the chain and resolves successfully, moving an enemy unit to that battlefield.

Sequence of Events:

1. Move: You move Irresistible Faefolk to an unoccupied battlefield. This triggers its ability: "When I move to a battlefield, you may move an enemy unit to that battlefield."
2. Chain State: The trigger is placed on the chain.
3. Reaction: Your opponent plays Gust in response, targeting the Irresistible Faefolk.
4. Chain Resolution (LIFO - Last-In, First-Out):
   • Resolution of Gust: Gust resolves first, returning the Irresistible Faefolk to its owner's hand.
   • Resolution of the Faefolk Trigger: Because the ability referenced "that battlefield" (where it originally moved to trigger), the ability successfully resolves. You may then move an enemy unit to that same battlefield.

According to Rule 359.3.f.3, information referenced from a trigger condition (such as "that battlefield") is checked when the trigger condition is fulfilled. Moving the source of the ability (the Faefolk) after the trigger condition is met does not invalidate the ability's target location.