One of the major themes of "There will come soft rains" is the idea of humans and science and their interaction as well as the conflict between them. Bradbury was working to present a picture of science as dangerous and destructive even as it has the veneer of doing a great service to humankind. The house is a symbol of this technology as it takes care of everything that people might need, the only problem being that the people are nothing more than blank shadows on the outside wall having been incinerated by nuclear weapons.
The way the house treats the dog serves only to emphasize this. The house knows to let the dog in, but the mechanical mice seem "annoyed" by the fact that they have to clean up after this dog, the dog that has come home to find its family gone. The dog is clearly horribly troubled by the journey it has taken home only to find its owners gone but the house is only an uncaring, robotic device. Once the dog has given in to despair and died, it incinerates it, just as the humans were incinerated by a different piece of technology.
In this way Bradbury draws another connection between technology or the products of science that are supposed to help humans that end up hurting them, further criticizing the work of scientists that pursue incredible means with which to more efficiently destroy humanity.