On Rosh Hashanah, we add Ya’aleh Ve’yabo in Birkat Ha’mazon, and recite the insert, "Et Yom Ha’zikaron Ha’zeh Et Yom Tob Mikra Kodesh Ha’zeh."
There is a debate among the Rishonim (Medieval Halachic scholars) as to whether one must repeat Birkat Ha’mazon if he forgot to add Ya’aleh Ve’yabo on Yom Tob. Many Rishonim, including the Rambam, Rosh, Ran and Ritba, maintain that one must repeat Birkat Ha’mazon in such a case, where one forgot Ya’aleh Ve’yabo in Birkat Ha’mazon on Yom Tob and did not realize his mistake until after he began the next Beracha ("Ha’tob Ve’ha’metib"). Others (Tosafot, Rashba), however, disagree. In their view, it is only on the first two nights of Pesah and of Sukkot, when there is an obligation to eat bread (or Masa, in the case of Pesah), that one must repeat Birkat Ha’mazon if he forgot Ya’aleh Ve’yabo. On all other days of Yom Tob, when there is an obligation to eat but no obligation to eat bread, one does not repeat Birkat Ha’mazon in such a case.
The Shulhan Aruch (Orah Haim 188:6) follows the first view, and rules that if one forgot to recite Ya’aleh Ve’yabo in Birkat Ha’mazon, and did not realize his mistake before beginning the next Beracha, he must repeat Birkat Ha’mazon. He does not distinguish between the first two days of Pesah and Sukkot and other days of Yom Tob, suggesting that this ruling applies to all Yamim Tobim.
As for the final Halacha, however, we employ the famous rule of "Safek Berachot Le’hakel" – that we do not recite a Beracha if there is some uncertainty as to whether it is required. In light of the debate among the Rishonim concerning this issue, one who forgot to add Ya’aleh Ve’yabo in Birkat Ha’mazon on Yom Tob does not repeat Birkat Ha’mazon, unless this occurred on the first two nights of Pesah or Sukkot. This is the ruling of the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Parashat Hukat (21), and of Hacham Bension Abba Shaul (Israel, 1923-1998), in Or Le’sion (vol. 3, 18:9).
Summary: If one forgot to add Ya’aleh Ve’yabo in Birkat Ha’mazon on Rosh Hashanah, or on other Yamim Tobim, he does not repeat Birkat Ha’mazon. The only exception to this rule is the first two nights of Pesah and Sukkot, when one who forgot to recite Ya’aleh Ve’yabo in Birkat Ha’mazon must repeat Birkat Ha’mazon.
