For example, a grave domain cleric gives an Iron Golem vulnerability to all damage, and someone else deals fire damage to the golem. Will the golem be healed for twice the amount it normally would be, or would the vulnerability not affect its absorption at all?
In the case of the Iron Golem, vulnerability is ignored.
The Iron Golem’s Fire Absorption trait says:
Whenever the golem is subjected to fire damage, it takes no damage and instead regains a number of hit points equal to the fire damage dealt.
The golem takes no fire damage. The rules for vulnerability state:
Resistance and then vulnerability are applied after all other modifiers to damage.
So vulnerability is applied after the feature that reduces damage to zero. So vulnerability is ignored. Instead the golem is healed by the amount of damage dealt, which is the unmodified damage. Since vulnerability is always going to be applied last after other features, like absorption, it is probable that vulnerability will never increase the healing gained by creatures with similar absorption features.
For a detailed exposition on the difference between damage taken and damage dealt, see this answer. This distinction is important here, because if damage taken and damage dealt are the same then the fire absorption trait doesn’t actually work.