If the barrier is on the forest and the dryad makes you move there, do you ignore the barrier and move there or stay in the same place?

You must remain on the space where you encountered the Dryad. You are not teleported to the Forest, as the Barrier Spell is in effect there.

See Page 15 of the main rulebook:

In any instance where a card's effect indicates that a character cannot perform an action or use an ability (such as casting a Spell or using an Object), the character cannot do so. In other words, the forbidding effects of cards override other abilities and effects.

Therefore the Barrier Spell overrides the effects of the Dryad Adventure Card!

Ell.