Can Azir's ability swap positions with a unit at Vilemaw's Lair, given that units cannot move from the Lair to base?

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Assuming Azir has an ability that attempts to swap two units by moving each to the other's location, you cannot move a unit off Vilemaw's Lair to a destination that the source of the Move forbids. If the Lair (or another effect) makes moving from that Lair to base illegal, the portion of the swap that would perform that illegal Move cannot be executed; any portions of the ability that can legally occur will still occur unless the ability is a single inseparable instruction that requires both Moves to execute together. Key rules: the Move source defines destination legality [444.1], and un-executable instructions are ignored or partially followed per [359.3.e.6] [359.3.e.11] [359.3.e.7].

If an attempt to swap with a unit at Vilemaw's Lair would require moving that unit to base but the Lair or another effect prohibits moving it to base, that Move is illegal and will not happen. Whether the other unit still moves depends on how Azir's ability is worded: if the ability contains separable instructions, the legal parts still resolve (partial execution) [359.3.e.11]; if the ability is a single instruction that cannot be executed because all of its targets or required Moves are illegal or unavailable, that instruction will not execute [359.3.e.7] [359.3.e.6]. The source of any Move governs destination legality [444.1].

Read the exact wording of Azir's ability to determine whether it issues separate Move instructions for each unit or a single combined instruction that requires both Moves to succeed. Check the source of the attempted Move for the unit at Vilemaw's Lair to see what destinations it allows; per [444.1], the Move source defines destination legality. If the Move from the Lair to base is illegal, mark that Move as unable to be executed. Apply [359.3.e.6] and [359.3.e.11]: ignore any instruction that can't be followed and partially execute the ability to the extent possible (e.g., move the other unit if that Move is legal and the ability's wording allows separable execution). If the ability is a single instruction that requires both Moves and all required Moves/targets are unavailable or illegal, treat the whole instruction as not executing per [359.3.e.7]. If any checks in the ability rely on information from a unit that became illegal/unavailable during resolution, treat that information as null per [359.3.e.12].

If another effect simultaneously grants permission to move the Lair unit to base (overriding the Lair prohibition), then the swap may be legal — because destination legality is defined by the Move source [444.1]. If Azir's ability uses targeting relationships (targets are chosen when the spell is played) and a unit is moved off-board (to a non-board zone) during resolution, that unit becomes a different object and any relationships to it are severed per the teachings in [FAQ:unleashed#22] and [359.3.e.12]. If the swap is implemented as a sequence where one Move must happen first and that first Move would leave an illegal location state for the second, the second Move's legality should be re-evaluated at its execution and follow the rules above about partial execution [359.3.e.11].

You asked whether Azir can swap positions with a unit at Vilemaw's Lair when units "cannot move from the Lair to base." The core points are: who defines whether a Move to a given Destination is legal, and what happens when a Move is illegal. The rules say the source of a Move defines destination legality [444.1]. So if the Lair (or some other active game effect) specifically prohibits moving a unit from that Lair to base, the Move instruction that would do that is illegal and therefore cannot be executed. The rules that govern what happens next are in the 359 series. Instructions that can't be followed because of illegal targets or other circumstances are ignored [359.3.e.6]. Instructions that can be partially followed are followed as much as possible and ignored otherwise [359.3.e.11]. If a whole instruction's targets are invalid or unavailable by the time the spell is finished being played, that instruction will not execute [359.3.e.7]. Applying those: if Azir's swap issues two separable Move instructions (one for each unit), and the Move that would take the Lair unit to base is illegal, that particular Move is ignored while the other Move may still happen if it is legal (partial execution) [359.3.e.11] [359.3.e.6]. If instead Azir's swap is worded as a single inseparable instruction that only succeeds if both Moves can occur, and the Lair-to-base Move is illegal, then the entire instruction will not execute [359.3.e.7]. Also be aware of interactions with targeting and objects that change zones: if a unit is moved to a non-board zone during resolution, it becomes a different object and targeting/relationships are severed (see [FAQ:unleashed#22] and the examples in [359.3.e.12]). In short: you cannot force a unit to move from the Lair to base if the Move source forbids that Destination [444.1]; the forbidden Move is ignored and the rest of the ability resolves only as allowed by the precise wording of Azir's ability and the partial/execution rules [359.3.e.6] [359.3.e.11] [359.3.e.7].