My opponent plays Promising Future. I reveal Brynhir and he reveals Dazzling Aurora. I play Brynhir first as mentioned in Promising Future. Does the Aurora get played?

Yes, Dazzling Aurora is played normally and enters the battlefield.

Here's why: When Promising Future resolves, "starting with the next player" (you), each player plays their banished card. You play Brynhir first, then your opponent plays Dazzling Aurora — both go onto the chain as pending items before Brynhir's "When you play me" trigger has a chance to do anything.

When these finalize, Brynhir resolves first (she was placed on the chain first), and her WYPM trigger goes on the chain above Dazzling Aurora. But Dazzling Aurora was already played and is a pending item on the chain — Brynhir's effect only prevents new card plays (taking cards from a zone and putting them on the chain). It doesn't stop cards already pending from resolving. So Dazzling Aurora finalizes and enters play as a Gear. (RiftJudge FAQ #229, #11227)

The practical consequence: Dazzling Aurora is on the board, but its "At the end of your turn" trigger (which would play a unit) is now affected by Brynhir's ongoing restriction — if it later triggers, the unit revealed by Aurora cannot be played because Brynhir's effect is active for the rest of the turn.