If, Heaven forbid, a person passes away during a holiday and is buried on Hol Ha’mo’ed, the formal Shiba mourning period does not begin until after the holiday. The family members are this in a kind of in-between status during Hol Ha’mo’ed, as they do not begin formal Abelut until after the holiday. However, despite the fact that Shiba does not begin during Hol Ha’mo’ed, certain aspects of mourning are observed already at that point. Specifically, family members observe mourning practices that apply in a private setting, but not those practices which take place in public. Thus, for example, as Rav Shemuel Pinhasi writes in his work Haim Va’hesed (p. 291), a mourner may not engage in marital relations on Hol Ha’mo’ed. As for Torah study, which is generally forbidden during Abelut, Hacham Ovadia Yosef rules (in Yabia Omer, vol. 4, Yoreh De’a 31:6) that a mourner may study Torah during Hol Ha’mo’ed. The mourner may recite Hallel with the congregation in the synagogue during Hol Ha’mo’ed, as people will notice if he refrains from reciting Hallel, and this would thus be considered public mourning, which is forbidden.
Rav Pinhasi notes that in such a case, when a mourner begins Abelut after the holiday, he must remember to wear Tefillin – with a Beracha – on the first day of mourning. In circumstances when the funeral is held on the same day as the family member’s passing, one does not wear Tefillin, but in this case, when Abelut is starting several days later, Tefillin is worn, and a Beracha is recited over the Tefillin.
An interesting question arises concerning the permissibility of work during Abelut in such a case. If a family member passes away during Hol Ha’mo’ed, the mourner could end up missing over ten days of work, since he does not work on Hal Ha’mo’ed and the last days of Yom Tob, and then he is not allowed to work during the seven days of Shiba. Rabbi Pinhasi writes (p. 291, Halacha 11) that in such a case, once the mourner had refrained from work for seven days, he may during Abelut have somebody else perform his work for him, such as by opening his shop. Thus, for example, if a person missed four days of work during Hol Ha’mo’ed, and then two days of Yom Tob, already the second day of Shiba he may have somebody do his work for him so that he does not suffer a serious financial loss.
Summary: If one’s family member passes away, Heaven forbid, during a holiday, formal Abelut begins only after the holiday. On Hol Ha’mo’ed, the mourner may not engage in marital relations, but he may study Torah and pray normally in the synagogue, including the recitation of Hallel. During Shiba in this case, he may have somebody perform his work for him – such as run his store – once he had already missed seven days of work because of his mourning observance.