If my opponent uses Gear Hate in response to me playing Runeblade Anvil, will Runeblade Anvil’s effect still resolve before it gets destroyed?

Gear Hate isn't a card I can find (no match in the card database), so I'll answer for any gear-killing card — and yes, Runeblade Anvil's effect resolves even if the Anvil is killed before that effect resolves.

Two things are happening separately. Playing the Anvil itself finalizes and resolves immediately with no reaction window (FAQ #8333); the window your opponent gets comes from the "When you play this" trigger, which goes on the chain and creates a Closed State. That ability doesn't reference the Anvil's location (no "here") or the Anvil itself, so removing it doesn't stop the banish-and-may-play — it resolves normally (FAQ #2006). Note the window is Closed, so only a [Reaction] card or ability can be used there; an [Action] spell like Detonate or Salvage can't be wedged in.

Also worth knowing: killing it triggers the second half of the card, "when this leaves the board," so you'd get two banish-and-play effects — the leaves-play trigger resolves first (newest on chain), then the play trigger. Their removal actually hands you a second card to play.