The accepted Halacha, based on the ruling of the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Parashat Hukat (21), is that if a person forgot to add "Ya’aleh Ve’yabo" in Birkat Ha’mazon on Yom Tob, he does not repeat Birkat Ha’mazon, unless this happened on one of the first nights of Pesach or one of the first nights of Sukkot. Although the Shulhan Aruch ruled that one repeats Birkat Ha’mazon in this case on any Yom Tob, Halacha in this instance follows the Ben Ish Hai’s ruling.
On Shabbat, however, if one forgot to add "Reseh" in Birkat Ha’mazon after one of the first two Shabbat meals (as opposed to Se’uda Shelishit), he must repeat Birkat Ha’mazon.
These two Halachot give rise to an interesting question in a case where Yom Tob falls on Shabbat, and after one of the first two meals one remembered to recite "Reseh" but forgot to recite "Ya’aleh Ve’yabo." We might have assumed that if this occurred on a Yom Tob other than the first nights of Pesach or the first nights of Sukkot, the individual does not need to repeat Birkat Ha’mazon. Since he recited "Reseh," and he mistakenly omitted only "Ya’aleh Ve’yabo" – the omission of which does not necessitate repeating Birkat Ha’mazon – he does not need to repeat the recitation.
However, both Hacham David Yosef (contemporary), in his Halacha Berura, and Rabbi Moshe Ha’levi (Israel, 1961-2000), in his Birkat Hashem, rule otherwise, and require the individual to repeat Birkat Ha’mazon in this case. The reason is that there are two Sefekot (uncertainties) at play in this case, each of which potentially requires repeating Birkat Ha’mazon. First, as mentioned, the Shulhan Aruch maintains that one must repeat Birkat Ha’mazon whenever one forgets to add "Ya’aleh Ve’yabo" on Yom Tob. Secondly, even according to the view of the Ben Ish Hai, it is possible that when Yom Tob falls on Shabbat, the status of "Ya’aleh Ve’yabo" rises to the status of "Reseh," such that its omission requires the repetition of Birkat Ha’mazon. Therefore, given these two uncertainties – and especially in light of the fact that one of these uncertainties arises due to the ruling of the Shulhan Aruch, whose positions are generally deemed authoritative – one must repeat Birkat Ha’mazon in this case.
Summary: When Yom Tob falls on Shabbat, and after one of the first two meals one remembered to recite "Reseh" in Birkat Ha’mazon but forgot to recite "Ya’aleh Ve’yabo," he must repeat Birkat Ha’mazon, regardless of which Yom Tob it is. This is despite the fact that when Yom Tob does not fall on Shabbat, and one forgot to recite "Ya’aleh Ve’yabo" in Birkat Ha’mazon, he repeats Birkat Ha’mazon only if this occurred on one of the first nights of Pesach or one of the first nights of Sukkot.
