The best example of symbolism is the house itself. It is the last remaining creation of humanity in this city, a place destroyed by nuclear weapons and almost entirely devoid of life. Since the house is automated and not equipped with artificial intelligence, it can only act on its programming, and can't make the decision to stop performing for the humans that are long-dead.
The sun came out from behind the rain. The house stood alone in a city of rubble and ashes. This was the one house left standing. At night the ruined city gave off a radioactive glow which could be seen for miles.
(Bradbury, "There Will Come Soft Rains," nexuslearning.net)
The house symbolizes order in the midst of chaos; it still acts and functions, and with everything around it destroyed. It is the last vestige of reason in the meaningless annihilation of nuclear war; it is also a place where there is still purpose, even if that purpose no longer has a larger meaning. The house fights entropy and ultimately fails, its rubble falling into the larger rubble of the destroyed city; this could symbolize a very pessimistic view of determination and the human struggle for meaning.