if i attack a battlefeild that has overzealous fan. but that battlefield is vilemaw lair. what happens? can he push me back to base?

No — the Fan's trigger still fires, but its effect whiffs: your unit stays at Vilemaw's Lair.

Overzealous Fan's ability is a move ("move an attacking unit to its base"), not a recall. Recalls are the only thing that bypass Vilemaw's Lair (Rules 455–456.3); "can't move from here to base" blocks every move instruction, including those from abilities (RiftJudge FAQ #4857). So the defending player can still activate the Fan (killing it as the upfront cost), but when the ability resolves, the move instruction is impossible and is ignored — your attacker remains at Vilemaw's Lair (Rule 359.3.e.6).

One real cost: the Fan dies either way, since killing it is the cost to even put the trigger on the chain.