If I have Lillia Fae Fawn at a battlefield with a trap card and next turn I move her back to base, would the trap card disappear or stay with the new sprite spawning from Lillia?

In Riftbound, there is no official card type or mechanic officially named a "trap." Assuming you are referring to a hidden card (a card placed facedown at a battlefield, as described in Rule 421.1), here is how your scenario resolves:

Short Answer

The hidden card stays. The hidden card is not affected by Lillia, Fae Fawn moving or by the Sprite token spawning.

Detailed Breakdown

1. Lillia's Movement: When Lillia moves from the battlefield to your base, her triggered ability ("When I move from a location, play a 3 [M] Sprite token with [Temporary] there") is placed on the chain.
2. The Hidden Card: Your hidden card is an independent game object in the Facedown Zone associated with that battlefield. It has no mechanical relationship to Lillia or the Sprite token.
3. The Sprite Token: When Lillia's triggered ability resolves, it plays the Sprite token at that battlefield. The battlefield now contains your hidden card, the new Sprite token, and any other units present.
4. Battlefield Control: Per FAQ #10369, you do not lose control of a battlefield just because Lillia left, provided there are other units present or the turn is in a "Closed State" (which occurs while items like Lillia's trigger are on the chain or resolving). Even if you were to lose control of the battlefield, you would only lose the hidden card during the next Cleanup, not immediately upon Lillia moving or the Sprite spawning.

In summary: The Sprite token spawning will not cause your hidden card to disappear. It will remain in the Facedown Zone associated with that battlefield.