Can Tideturner switch locations with a unit at Vilemaw's Lair?

No — you cannot swap a unit out of Vilemaw's Lair via Tideturner if Tideturner was played to base.

Tideturner's "When you play me" ability uses the word "move." Vilemaw's Lair says "Units can't move from here to base." Since a "can't" restriction beats a "can" effect, the targeted unit in the Lair cannot move to base.

Here's what happens:

1. Tideturner is played to base and targets a unit in Vilemaw's Lair.
2. Tideturner successfully moves to the Lair.
3. The targeted unit cannot move to base — it stays in Vilemaw's Lair.

The swap partially resolves: Tideturner ends up in the Lair alongside the other unit. (FAQ #7241, #8637)

One exception: If Tideturner's original location is a different battlefield (not your base), the swap completes fully — Vilemaw's Lair only blocks movement to base, not to another battlefield. (FAQ #11402)