Making vows is to be discouraged for multiple reasons. When it comes to Birkath HaMazon, vows may have certain ramifications.
If three people who vowed not to have any benefit from each other, eat a meal with bread together, they cannot join together for Zimmun (invitation to recite Birkath HaMazon). This only applies, however, when each one is eating from his own loaf. If, on the other hand, they all eat from the loaf of the householder, then they do join together, because they are all eating from one loaf.
In a case where Reuben swore that he would not speak to Shim'on, and the two of them ate a meal with bread together with a third person, Reuben may not be the one to recite the Zimmun, because he is inviting both Shim'on and the third person to recite Birkath HaMazon and he vowed not to speak to Shim'on. Obviously Shim'on or the third person, neither of whom swore not to speak to another, may recite the Zimmun.
(See Shulhan 'Arukh with Rama, 196:3. Kaf Hahayyim, ibid., Oth 18. Mishnah Berurah, ibid., 9)
zimun and bircat hamazon, brikath hammazon, birkath hamazon