Halacha forbids observing a fast on any of the eight days of Chanukah, with the exception of a Ta'anit Halom, meaning, a fast observed in response to a inauspicious dream. If one feels very disturbed by a bad dream, he may fast that day (that is, from the morning after experiencing this dream), even if it is Chanukah. However, given the impropriety of fasting during Chanukah, he must then observe another fast after Chanukah to atone for having fasted on Chanukah. This applies to Shabbat, as well: one who dreams a frightening dream on Friday night and is disturbed by it may fast that Shabbat, but he must then observe another fast to atone for having fasted on Shabbat.
People who follow the laudable practice which is recorded by Rav Pinchasi, in his work "Chayim Va'chesed" to fast on the Yartzheit of a parent may not do so if the Yartzheit falls during Chanukah. They must instead observe the fast either before or after Chanukah.
One may not deliver a eulogy during Chanukah, except at the funeral of a Torah scholar. The eulogies customarily delivered at the conclusion of the seven-day and thirty-day mourning periods, or on the Yartzheit, are forbidden during Chanukah. If these ceremonies are held on Chanukah, the rabbis and other speakers should limit themselves to words of Torah, and must not eulogize the deceased individual.
Visits to relatives' gravesites are also forbidden during Chanukah.
All these prohibitions apply only on the actual eight days of Chanukah. On the day prior to Chanukah and on the day immediately following Chanukah, one may observe a fast, deliver a eulogy and visit a cemetery. There is no concept of "Isru Chag" (a quasi festival observed on the day following a festival) with respect to Chanukah as there is regarding other festivals.
All laws of mourning apply on Chanukah. Unlike most festivals, which suspend or cancel the observance of mourning, Chanukah does not interfere at all with any of the traditional mourning practices. Keri'a (rending garments) and all other observances related to mourning, including the recitation of Kadish, apply on Chanukah just as on any other day.
