The Shulhan Aruch (Orah Haim 188:5) addresses the case of a person who was reciting Birkat Ha’mazon on Yom Tob, and immediately after reciting "Boneh Yerushalayim" he realized that he had forgotten to insert "Ya’aleh Ve’yabo," as is required on Yom Tob. In such a case, the Shulhan Aruch rules, the individual recites at that point a special Beracha that was instituted by the Sages specifically for this case, and which appears in some Siddurim: "Baruch Ata Hashem Asher Natan Yamim Tobim Le’Yisrael Le’sasson U’le’simha Et Yom…Ha’zeh, Baruch Ata Hashem Mekadesh Yisrael Ve’ha’zmanim." The Shulhan Aruch notes that even if one had begun reciting the next Beracha of Birkat Ha’mazon before he realized his mistake, he may recite the Beracha of "Asher Natan," unless he had already reached the word "La’ad." At that point, the Shulhan Aruch writes, one can no longer recite "Asher Natan," and so he must return to the beginning of Birkat Ha’mazon.
The clear implication of this final ruling is that one who forgot to recite "Ya’aleh Ve’yabo" in Birkat Ha’mazon on Yom Tob must repeat Birkat Ha’mazon. Maran here states that once a person reached "La’ad," his only option is to return to the beginning of Birkat Ha’mazon – implying that the mistaken omission of "Ya’aleh Ve’yabo" on Yom Tob invalidates Birkat Ha’mazon, requiring one to recite it again.
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), however, rules differently, noting the position of the Rashba (Rav Shlomo Ben Aderet of Barcelona, 1235-1310) distinguishing in this regard between different occasions. The Rashba maintained that on the first nights of Pesach and the first nights of Sukkot, when eating a meal is strictly required, one who omits "Ya’aleh Ve’yabo" must repeat Birkat Ha’mazon. On all other days of Yom Tob, however, according to the Rashba, one who forgot to recite "Ya’aleh Ve’yabo" in Birkat Ha’mazon has nevertheless fulfilled his obligation, and is not required to repeat Birkat Ha’mazon. In light of the Rashba’s opinion, this situation qualifies as one of a "Safek Berachot" – where there is uncertainty surrounding one’s requirement to recite Berachot. Accordingly, we apply the rule of "Safek Berachot Le’hakel," which says that when in doubt we do not recite a Beracha. Although the Shulhan Aruch clearly requires one to repeat Birkat Ha’mazon if he forgot to add "Ya’aleh Ve’yabo" on any Yom Tob, the view of the Rashba makes the obligation to repeat Birkat Ha’mazon questionable in such a situation.
This is the opinion accepted by Hacham Ovadia Yosef, both in Halichot Olam and in Hazon Ovadia. He explains that although Birkat Ha’mazon constitutes a Biblical requirement, such that one might have assumed that in this situation of doubt one must repeat Birkat Ha’mazon (as we always follow the stringent possibility in situations of doubt relevant to a Torah obligation), nevertheless, the recitation of "Ya’aleh Ve’yabo" is required only Mi’de’rabbanan (by force of Rabbinic enactment). Therefore, if one recited Birkat Ha’mazon on Yom Tob but mistakenly omitted "Ya’aleh Ve’yabo," he has fulfilled the Biblical obligation of Birkat Ha’mazon, and the uncertainty revolves around the Rabbinic obligation of "Ya’aleh Ve’yabo." Therefore, given the uncertainty, he should not repeat Birkat Ha’mazon.
According to the final Halacha, then, one who forgot to recite "Ya’aleh Ve’yabo" in Birkat Ha’mazon on the first nights of Pesach or the first nights of Sukkot should repeat Birkat Ha’mazon, but if this happened on a different Yom Tob, one does not repeat Birkat Ha’mazon. On any Yom Tob, if one realizes his mistake before reciting the word "La’ad" in the next Beracha, he inserts the special Beracha of "Asher Natan Yamim Tobim Le’Yisrael."
Summary: If one forgot to add "Ya’aleh Ve’yabo" in Birkat Ha’mazon on Yom Tob, and he realized his mistake after "Boneh Yerushalayim," then as long as he had yet to reach the word "La’ad" in the next Beracha, he should recite at that point the special Beracha of "Asher Natan Yamim Tobim Le’Yisrael" which appears in some editions of the Siddur. If he realized his mistake after reciting "La’ad," then he simply continues reciting Birkat Ha’mazon, unless this happened on the first nights of Pesach or the first nights of Sukkot, in which case he must start Birkat Ha’mazon again from the beginning.
