Can my opponent use Overzealous ability if I move up with Ezreal dashing?

Yes, your opponent can use Overzealous Fan's ability when Ezreal, Dashing moves into the battlefield to attack, but the timing of the triggers is critical.

The Ruling

Both Ezreal, Dashing and Overzealous Fan have triggered abilities that occur when they gain their combat designations (Attacker and Defender). Because it is your turn (you are the one moving Ezreal to attack), your "When I attack" trigger goes on the chain first, followed by your opponent's "When I defend" trigger.

Because the chain resolves in LIFO (Last-In, First-Out) order, the Overzealous Fan's ability will resolve before Ezreal deals his ability damage.

The Sequence

1. Ezreal moves to the battlefield: He becomes the Attacker. Overzealous Fan becomes the Defender.
2. Triggers go on the Chain:
   • Ezreal's "When I attack" trigger is placed on the chain first (Active Player).
   • Overzealous Fan's "When I defend" trigger is placed on the chain second (Non-Active Player).
3. Resolution (LIFO):
   • Overzealous Fan's ability resolves first: The opponent chooses whether to kill the Fan. If they do, Ezreal is moved back to your base immediately.
   • Ezreal's ability attempts to resolve: Because Ezreal is no longer "here" (he is now at the base), his ability will "whiff" (fail to resolve) because he has lost his location and cannot find a target "here" (See Rule 356.3.e.12).

Summary

If your opponent chooses to sacrifice the Fan, Ezreal will be sent back to your base and will deal no damage.

Note: According to the official FAQ, the opponent must choose Ezreal as the target when the ability is put on the chain, but they don't make the final decision to "kill" the Fan until the moment the ability actually resolves.