Objection 1. It would seem that any priest can absolve his subject from excommunication. For the chains of sin are stronger than those of excommunication. But any priest can absolve his subject from sin. Therefore much more can he absolve him from excommunication.
Objection 2. Further, if the cause is removed the effect is removed. But the cause of excommunication is a mortal sin. Therefore since any priest can absolve (his subject) from that mortal sin, he is able likewise to absolve him from the excommunication.
On the contrary, It belongs to the same power to excommunicate as to absolve from excommunication. But priests of inferior degree cannot excommunicate their subjects. Neither, therefore, can they absolve them.
I answer that, Anyone can absolve from minor excommunication who can absolve from the sin of participation in the sin of another. But in the case of a major excommunication, this is pronounced either by a judge, and then he who pronounced sentence or his superior can absolve--or it is pronounced by law, and then the bishop or even a priest can absolve except in the six cases which the Pope, who is the maker of laws, reserves to himself: the first is the case of a man who lays hands on a cleric or a religious; the second is of one who breaks into a church and is denounced for so doing; the third is of the man who sets fire to a church and is denounced for the deed; the fourth is of one who knowingly communicates in the Divine worship with those whom the Pope has excommunicated by name; the fifth is the case of one who tampers with the letters of the Holy See; the sixth is the case of one who communicates in a crime of one who is excommunicated. For he should not be absolved except by the person who excommunicated him, even though he be not subject to him, unless, by reason of the difficulty of appearing before him, he be absolved by the bishop or by his own priest, after binding himself by oath to submit to the command of the judge who pronounced the excommunication on him.
There are however eight exceptions to the first case: (1) In the hour of death, when a person can be absolved by any priest from any excommunication; (2) if the striker be the doorkeeper of a man in authority, and the blow be given neither through hatred nor of set purpose; (3) if the striker be a woman; (4) if the striker be a servant, whose master is not at fault and would suffer from his absence; (5) if a religious strike a religious, unless he strike him very grievously; (6) if the striker be a poor man; (7) if he be a minor*, an old man, or an invalid; (8) if there be a deadly feud between them.
There are, besides, seven cases in which the person who strikes a cleric does not incur excommunication: (1) if he do it for the sake of discipline, as a teacher or a superior; (2) if it be done for fun; (3) if the striker find the cleric behaving with impropriety towards his wife his mother, his sister or his daughter; (4) if he return blow for blow at once; (5) if the striker be not aware that he is striking a cleric; (6) if the latter be guilty of apostasy after the triple admonition; (7) if the cleric exercise an act which is altogether contrary to the clerical life, e.g. if he become a soldier, or if he be guilty of bigamy [Namely, that which is known by canonists as "similar bigamy"].
Reply to Objection 1. Although the chains of sin are in themselves greater than those of excommunication, yet in a certain respect the chains of excommunication are greater, inasmuch as they bind a man not only in the sight of God, but also in the eye of the Church. Hence absolution from excommunication requires jurisdiction in the external forum, whereas absolution from sin does not. Nor is there need of giving one's word by oath, as in the case of absolution from excommunication, because, as the Apostle declares (Hebrews 6:16), controversies between men are decided by oath.
Reply to Objection 2. As an excommunicated person has no share in the sacraments of the Church, a priest cannot absolve him from his guilt, unless he be first absolved from excommunication.
Source: Summa Theologica (New Advent)