When someone passes away on Ḥol Ha-mo’ed, it is permissible to take care of all burial needs. This includes sewing the shrouds and digging the grave, when necessary (SA 547:10; 12:11 below). However, eulogies are not delivered on Ḥol Ha-mo’ed, since there is a Torah obligation to rejoice then. For this reason, one should make sure not to indulge in excessive crying and mourning (MK 27a; SA 547:1).
If the deceased is a Torah scholar, eulogies are delivered at the funeral, as the honor due to Torah overrides the joy of the festival (MK 27b; SA YD 401:5). Some maintain that nowadays nobody has mastered the entire Torah, and thus even Torah scholars should not be eulogized on the festival (MA; MB 547:12). In practice, the custom is to eulogize a great Torah scholar who is well known as an educator (marbitz Torah) or halakhic authority (moreh hora’ah la-rabim), but to make the eulogies shorter than they otherwise would have been.
According to Shulḥan Arukh, even on Ḥol Ha-mo’ed, relatives who are obligated to mourn tear their clothes during the funeral (SA 547:6). Nevertheless, many people (both Sephardim and Ashkenazim) tear only for their parents, not for other relatives (Rema ad loc.; Kaf Ha-ḥayim ad loc. 30).
After the funeral, the mourners resume wearing their festival clothes. They do not begin mourning practices, since the joy of the festival defers the mourning. Only after the festival do the relatives sit shiva. Even though the mourners do not sit shiva on the festival, their close friends come to visit and comfort them (SA 548:6).
The Sages ordained that on a festival, a Kohen should not examine someone with symptoms of tzara’at, because if the Kohen determines that the person is impure, it would ruin his festival. Rather, the examination should take place after the festival (MK 7a; MT, Laws of Yom Tov 7:16).
Fasting is forbidden on the festival, even personal fasts that one might undertake as atonement for his sins (SAH 288:3; MB 529:1).
As described above, on Ḥol Ha-mo’ed we should avoid anything that causes distress, even mitzvot such as delivering eulogies and examining tzara’at. How much more so must we be careful to avoid conversations likely to cause distress. This includes speaking about loved ones who have passed on, or about aggravating subjects (Ru’aḥ Ḥayim 529:4).