The main conflict expressed in the story has to do with the fate of mankind. While the world lays in ruins outside the house, with the outlines of the former occupants burned into its side, Bradbury makes a point of focusing on the death and destruction of people, while the house, still remains.
He is making a larger comment on the potential future that humanity faces, the advanced technology is both a great invention and a great menace. Clearly, the same society that created the smart house that takes care of everything is also responsible for the destruction of all the humanity.
Even though the house is so efficient, it cannot help the starving dog survive, he becomes a victim of the same technology that has made the former occupants lives so easy.
This is an interesting question, as there really isn't a "character" to experience conflict in the story. However, if I had to guess, I'd consider the conflict to be the house's inability to sustain itself without human oversight. While things go on normally for a certain amount of time, such as the preparation and serving of meals, daily cleaning and maintenance, and even the reading of poetry in the evenings (hence the title of the short story), it is unable to stop the fire.
Once the fire starts, the sprinklers and "robot mice" attempt to put it out, but without any outside assistance, the house can only hold out so long before it succumbs to the flames. It's a difficult story to understand at some points, but at the same time, it shows that things not built by human hands (nature itself, wildlife, etc.) are able to survive just fine without our protection and intervention. This is seen most notably in the poem by Sara Tisdale, included in the short story.