Do people have to take time off work during the 7 days of shiva?
An aveil (mourner) may not go to work for the duration of the seven days. This is a rabbinic prohibition, whose purpose is that the mourner should be able to focus on the mourning.
Sources: Yoreh De’ah 380:1, and commentaries. The first three days of mourning are the most strict; past this, a poor person, who has nothing to eat, may work from home (380:2).