The Shulhan Aruch devotes a lengthy chapter (Orah Haim 548) to the subject of mourning during a holiday. He writes that if a person passes away, G-d forbid, on Yom Tob or Hol Ha’mo’ed, the family members do not begin observing Shiba until after the holiday. As the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) explains, mourning is not observed during Yom Tob because the public Misva of Simhat Ha’regel (rejoicing on the festival) overrides the private obligation of Abelut (mourning).
In such a case, the thirty days of Sheloshim – the prohibitions of shaving, haircutting, etc. – are counted from the day of the burial. Although actual mourning does not begin until after the holiday, the interim days count toward the period of thirty days, starting from the day of the burial. Moreover, in the Diaspora, where two days of Yom Tob are observed, the final day of the Yom Tob counts as the first day of Shiba, even though mourning is not observed on that day. For example, if somebody passed away during this past Pesah (5774), which concluded with the final days on Monday and Tuesday, the mourners would begin observing Shiba on Tuesday night, but the final day of Shiba would be Monday. Since the additional day of Yom Tob is observed only Mi’de’rabbanan (by force of Rabbinic enactment), it may count towards Shiba, even though mourning is not yet observed on that day.
If somebody passed away just before Yom Tob, and the family members observed some mourning before the onset of the holiday, the Shiba period is discontinued. Even if the mourners observed Shiba for just one hour on Ereb Yom Tob, the onset of the holiday ends Shiba, and they do not continue Shiba after Yom Tob. The Sheloshim observance resumes after the holiday, but the brief period of Shiba observed beforehand counts as seven days with regard to the Sheloshim observance, and the days of the holiday also count towards Sheloshim. Thus, for example, if a person sat Shiba for a brief period on Ereb Pesah, after Pesah he observes just fifteen days of Sheloshim, as the period before Pesah counts as seven days, such that after the eight days of the holiday only fifteen days remain.
If a person sat Shiba for a brief period before Sukkot, then he observes only eight days of Sheloshim after the holiday. The brief period of Shiba counts as seven days, the seven days of Sukkot make it fourteen days, and Shemini Aseret, which is a separate holiday, counts as seven days, bringing the total to 21. And thus after Simhat Torah, he has only eight days remaining.
There is a debate among the Halachic authorities as to how the Sheloshim period is calculated when a person passes away during Sukkot. The Mishna Berura rules that although Shemini Aseret is normally regarded as a separate holiday, in this case it counts as only one day of Sheloshim. Hence, each day since the burial counts as just one day. This is the view of the Hagahot Maimoniyot, cited by Maran in Bet Yosef. Hacham Ovadia Yosef, however, in his Yabia Omer (vol. 9, Y.D. 47), disagrees. He notes that although Maran cited this view in Bet Yosef, he does not mention it in Shulhan Aruch. And since there is a general rule of "Halacha Ke’dibreh Ha’mekel Be’abel" – Halacha follows the lenient position when the authorities disagree on matters relevant to Abelut – we may follow the lenient ruling, that Shemini Aseret counts as seven days even when the burial took place during Sukkot. This ruling also appears in Hazon Ovadia – Abelut (vol. 3, p. 117).
Summary: If a person passes away during Yom Tob or Hol Ha’mo’ed, Shiba begins after the holiday. The interim days count toward the thirty days of Sheloshim, and in the Diaspora, the final day of Yom Tob counts as the first day of Shiba. If the person passed away during Sukkot, the day of Shemini Aseret counts as seven days of Sheloshim. If a person passed away just before Yom Tob, and the family members sat Shiba even for just a brief period before the holiday, Shiba does not resume after the holiday. The brief Shiba period which they observed counts as seven days of Sheloshim, and each day during the holiday counts as one day of Sheloshim. If this happened before Sukkot, then Shemini Aseret counts as seven days of Sheloshim.
