Why does the house keep going, without human occupants, in "There Will Come Soft Rains"?
The house keeps going because it has been programmed, and as long as the programming does not change it will continue.
In “There Will Come Soft Rains” Bradbury tells us a story of a house that continues to care for its inhabitants long after they become “the five spots of paint” (p. 2, see first link). The house is set on automatic, and does not respond to the fact that no one is responding to it. Clearly, the house demonstrates advanced technology.
Until this day, how well the house had kept its peace. How carefully it had inquired, "Who goes there? What's the password?" and, getting no answer from lonely foxes and whining cats, it had shut up its windows and drawn shades in an old-maidenly preoccupation with self—protection which bordered on mechanical paranoia. (p. 2, see first link).
In the story, humans have created enough technology to make their world easier, but the technology has destroyed them. The house hopelessly wants its inhabitants to respond, but they cannot because they are dead.