The house left standing after a nuclear war symbolizes the human worship of technology. In the story, the house is pictured as a god. For example, Bradbury writes:
The house was an altar with ten thousand attendants, big, small, servicing, attending, in choirs
an incinerator which sat like evil Baal in a dark corner.
Likening the incinerator to "Baal" reinforces the house as symbol of idolatry, for Baal was a false, impotent god who could not conjure up fire in the Biblical story of Elijah and the priests of Baal.
The frightened, emaciated dog covered in sores symbolizes life: life that is sick and dying. The false god of the house doesn't serve this life but simply waits for it to die. The house sees life, the living organism, as a problem, a mess that it doesn't want to deal with.
[The dog] moved in and through the house, tracking mud. Behind it whirred angry mice, angry at having to pick up mud, angry at inconvenience.
Having the mechanical cleaning devices shaped to look like mice and rats symbolizes that they are pests, invaders, underlining the ambivalence of the mechanical as "helpful" to humans and other living things. Do we really want mice and rats in our home, even if they are supposed to be useful? Isn't reliance on them a little disgusting?
The Sara Teasdale poem which lends the story its name is read by the house, which is incapable of understanding the poem's irony. The poem symbolizes the victory of nature over the machine.
Finally, the fire at the end symbolizes the power of nature: the house, as powerless as Baal, is unable to defend itself against this force and so is destroyed.