Does Azir Ascendants move exhaust the unit it swaps with?

No, Azir, Ascendant's ability does not exhaust the unit it swaps with.

According to the RiftJudge FAQ (ID 7964), spells and abilities that move units do not change their exhaust/ready position unless they specifically state otherwise. Since Azir, Ascendant's ability does not mention exhausting the target unit, that unit remains in its current state (ready or exhausted) after the swap.