This expression usually means to look after domestic activities and obligations, especially during the absence of one's mate.
In this story, the expression is used ironically in that the firemen on Montag and Beatty's squad light fires intentionally. They do this as part of the government's effort to eradicate books and their provocative content. This massive bookburning rings a bell with similar measures taken during the Inquisition, the spread of the Nazi regime in Europe as well as the "witch hunts" of the MaCarthy trials in the United States. (Note that several of Bradbury's personal friends fell victim to the latter.) So the call to duty in for destruction rather than for maintaining domestic peace and order.
Bookburining is not the only strategy used by the government to subdue the people. Real families are replaced by "interactive" virtual ones (holograms) and citizens regularly dope themselves up to escape the hard reality of their meagre existence. (In fact, the suicide level is so high that there are technicians on permanent call just to pump people's stomachs or to get rid of the bodies, as the case may be.) Mildred's permanent symbiosis with her "seashell" is frightfully akin to our "plugged in" generation of Ipods, Smartphones, and Blackberries.
The scheme is to offer the people "bread and circuses" as a diversion to keep them from confronting and dealing with their real problems instead. Such is the topsy-turvy dystopian world of Bradbury's Fahrenheit 451, which is reminiscient of those found in Orwell's 1984 and Huxley's Brave New World, among others.