Does the son's mother become forbidden to the father if the father allows his son to marry the girl he loves?

The ruling on an oath to declare one's wife unlawful depends on the husband's intention. If he intended divorce, then it is a divorce. If he intended ẓihār (a form of injurious oath likening one's wife to a prohibited relative), then it is ẓihār. If he intended an oath or had no specific intention, then it is an oath.

- If the father, by declaring his wife unlawful, intended divorce: If he allowed his son to marry, his divorce takes effect, and he may take back his wife within her ʿiddah (waiting period) if the divorce was less than three pronouncements. - If he intended ẓihār: Ẓihār takes effect, and it becomes unlawful for him to have intercourse with his wife until he performs the ẓihār expiation mentioned in the Quran: freeing a slave, or fasting for two consecutive months, or feeding sixty needy persons. - If he intended neither divorce nor ẓihār: He must pay the expiation for an oath, which is feeding ten needy persons or clothing them. - If the father did not allow his son to marry: Then there is no expiation for him, and no divorce, ẓihār, or oath takes effect.

The lawful oath is an oath by Allah Almighty, and an oath to declare one's wife unlawful is not lawful.