If my opponent plays Tideturner at his base but he has a unit on Vilesaw's Lair, how does the interaction work?

The interaction between Tideturner and Vilemaw's Lair is governed by the fact that Vilemaw's Lair contains a "can't" restriction, which overrides the "can" effect of Tideturner's ability.

According to RiftJudge FAQ #7241:

1. The Play: When your opponent plays Tideturner at their base and targets a unit in Vilemaw's Lair, the ability attempts to resolve.
2. Partial Resolution: Because Tideturner's ability is a "move" effect, it is blocked by Vilemaw's Lair's text: "Units can't move from here to base."
3. The Outcome:
   • Tideturner successfully moves from the base to Vilemaw's Lair.
   • The targeted unit in Vilemaw's Lair cannot move to the base due to the battlefield's restriction.
   • The targeted unit remains in Vilemaw's Lair.

In summary, the swap fails to complete fully; Tideturner ends up in the Lair with the other unit, but the unit in the Lair is unable to leave.