My opponent has a Forge Fire Cape at a battlefield. I move a five might recruit token to the battlefield and use Azir Ascendant action to swap the units. Does Azir take 2 damage?

No, Azir, Ascendant does not take 2 damage in this scenario.

Here is the breakdown of why:

1. Forgefire Cape's Trigger: According to the card text retrieved, Forgefire Cape's effect is: "When I attack or defend, deal 2 to all enemy units here."
2. Trigger Condition: This ability only triggers when the unit equipped with the Forgefire Cape is designated as an attacker or defender during combat.
3. The Interaction: Moving a unit to a battlefield or swapping units using Azir, Ascendant's ability does not initiate combat, nor does it designate any unit as an attacker or defender. Because no combat is occurring and no unit has been designated as an attacker or defender, the Forgefire Cape's ability does not trigger.

Even if combat were occurring, the Forgefire Cape's effect only triggers when the unit it is attached to attacks or defends. Simply moving or swapping units into the same battlefield as the Forgefire Cape does not cause it to deal damage.