Objection 1. It would seem that no one can dispense from an oath. Just as truth is required for a declaratory oath, which is about the past or the present, so too is it required for a promissory oath, which is about the future. Now no one can dispense a man from swearing to the truth about present or past things. Therefore neither can anyone dispense a man from making truth that which he has promised by oath to do in the future.
Objection 2. Further, a promissory oath is used for the benefit of the person to whom the promise is made. But, apparently, he cannot release the other from his oath, since it would be contrary to the reverence of God. Much less therefore can a dispensation from this oath be granted by anyone.
Objection 3. Further, any bishop can grant a dispensation from a vow, except certain vows reserved to the Pope alone, as stated above (88, 12, ad 3). Therefore in like manner, if an oath admits of dispensation, any bishop can dispense from an oath. And yet seemingly this is to be against the law [Caus. XV, qu. 6, can. Auctoritatem, seqq.: Cap. Si vero, de Jurejurando]. Therefore it would seem that an oath does not admit of dispensation.
On the contrary, A vow is more binding than an oath, as stated above (Article 8). But a vow admits of dispensation and therefore an oath does also.
I answer that, As stated above (Question 88, Article 10), the necessity of a dispensation both from the law and from a vow arises from the fact that something which is useful and morally good in itself and considered in general, may be morally evil and hurtful in respect of some particular emergency: and such a case comes under neither law nor vow. Now anything morally evil or hurtful is incompatible with the matter of an oath: for if it be morally evil it is opposed to justice, and if it be hurtful it is contrary to judgment. Therefore an oath likewise admits of dispensation.
Reply to Objection 1. A dispensation from an oath does not imply a permission to do anything against the oath: for this is impossible, since the keeping of an oath comes under a Divine precept, which does not admit of dispensation: but it implies that what hitherto came under an oath no longer comes under it, as not being due matter for an oath, just as we have said with regard to vows (88, 10, ad 2). Now the matter of a declaratory oath, which is about something past or present, has already acquired a certain necessity, and has become unchangeable, wherefore the dispensation will regard not the matter but the act itself of the oath: so that such a dispensation would be directly contrary to the Divine precept. On the other hand, the matter of a promissory oath is something future, which admits of change, so that, to wit, in certain emergencies, it may be unlawful or hurtful, and consequently undue matter for an oath. Therefore a promissory oath admits of dispensation, since such dispensation regards the matter of an oath, and is not contrary to the Divine precept about the keeping of oaths.
Reply to Objection 2. One man may promise something under oath to another in two ways. First, when he promises something for his benefit: for instance, if he promise to serve him, or to give him money: and from such a promise he can be released by the person to whom he made it: for he is understood to have already kept his promise to him when he acts towards him according to his will. Secondly, one man promises another something pertaining to God's honor or to the benefit of others: for instance, if a man promise another under oath that he will enter religion, or perform some act of kindness. On this case the person to whom the promise is made cannot release him that made the promise, because it was made principally not to him but to God: unless perchance it included some condition, for instance, "provided he give his consent" or some such like condition.
Reply to Objection 3. Sometimes that which is made the matter of a promissory oath is manifestly opposed to justice, either because it is a sin, as when a man swears to commit a murder, or because it is an obstacle to a greater good, as when a man swears not to enter a religion: and such an oath requires no dispensation. But in the former case a man is bound not to keep such an oath, while in the latter it is lawful for him to keep or not to keep the oath, as stated above (07, ad 2). Sometimes what is promised on oath is doubtfully right or wrong, useful or harmful, either in itself or under the circumstance. On this case any bishop can dispense. Sometimes, however, that which is promised under oath is manifestly lawful and beneficial. An oath of this kind seemingly admits not of dispensation but of commutation, when there occurs something better to be done for the common good, in which case the matter would seem to belong chiefly to the power of the Pope, who has charge over the whole Church; and even of absolute relaxation, for this too belongs in general to the Pope in all matters regarding the administration of things ecclesiastical. Thus it is competent to any man to cancel an oath made by one of his subjects in matters that come under his authority: for instance, a father may annul his daughter's oath, and a husband his wife's (Numbers 30:6, seqq.), as stated above with regard to vows (88, A8,9).
Source: Summa Theologica (New Advent)