If I have the battlefield Vilemaw's Lair and my opponent uses The Syren, does the battlefield override the gear?

The movement effect of The Syren will fail to move your unit if it is located at Vilemaw's Lair.

Here is the breakdown of why the battlefield’s restriction takes precedence:

1. The Conflict: Vilemaw's Lair explicitly states that "Units can't move from here to base." The Syren has an activated ability that instructs you to "Move a friendly unit at a battlefield to its base."
2. "Can't" Overrides "Can": According to the Riftbound "Silver Rule" established in the October 2025 patch notes, "Can't beats can." A hard restriction on a battlefield ("Units can't move...") overrides the effect of a card that attempts to perform that action.
3. Recall vs. Move: It is important to distinguish between "moving" and "recalling." While Vilemaw's Lair blocks moves, it does not block recalls (such as the end-of-combat cleanup or effects like Zhonya's Hourglass). Since The Syren uses the specific game action "Move," it is subject to the battlefield's restriction.

In summary, the restriction on Vilemaw's Lair acts as a "can't" rule that prevents The Syren's movement effect from successfully resolving.