The Shulhan Aruch (Orah Haim 341) rules that if one had made a vow that interferes with his enjoyment of Shabbat, he may perform Hatarat Nedarim – the annulment of the vow – on Shabbat. An example would be if he vowed not to eat, or even not to eat meat or not to drink wine. Since there is a Misva to enjoy fine food and drink on Shabbat, one may have such a vow annulled on Shabbat so that he can enjoy Shabbat properly. The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) adds that this applies also to a vow that one will refrain from sleep, or refrain from wearing his fine clothes. One may have these vows annulled on Shabbat so he can sleep and wear his fine clothes.
However, vows that have no impact on one’s enjoyment of Shabbat may not be annulled on Shabbat, as this entails an unnecessary hassle on Shabbat. One who wishes to have such vows annulled must wait until after Shabbat. However, if a husband hears on Shabbat about a vow made by his wife, he may annul the vow on Shabbat, since a husband has the authority to annul a wife’s vows only on the day he hears about it.
Summary: One may have a vow annulled on Shabbat if it would interfere with his enjoyment of Shabbat, but otherwise, he should wait until after Shabbat to have it annulled.