What overall effect does the author's personification of the house have on the readers as the house burns down?

This is an interesting question, as the response a reader has to the house is complicated; the house is both personified, and yet, at the same time, its mechanical, mindless nature is emphasized as it goes about the business of caring for the family, a job which is now pointless because the family is dead. Nevertheless, "overall" is the key word, and I would say that the overall effect of the personification is one of sadness and horror. The death of the family from, presumably, the fire of a nuclear attack—we know it is a nuclear attack because the image of the family burned into the side of the house is borrowed straight from John Hersey's Hiroshima—happens before the story opens. The fiery end of the house, though somewhat different from what must have been the instant death of the family, nevertheless imitates the death of the family and allows us to mourn. This is especially so as Bradbury makes the house seem more human than ever by heightening its personification at the end of the story.
The personification of the house which makes us feel sad is reinforced by the personification of the fire. The fire is seen as a psychopath that the house must battle:

The fire crackled up the stairs. It fed upon Picassos and Matisses in the upper halls, like delicacies, baking off the oily flesh, tenderly crisping the canvases into black shavings.

Nevertheless, because the house continues to be personified, we feel for its courage in trying to fight the fire and the pain and grief of its defeat. At the end, it is depicted as far more human than mechanical. It seems to feel pain, fear, desperation, and a sense of tragedy:

The house shuddered, oak bone on bone, its bared skeleton cringing from the heat, its wire, its nerves revealed as if a surgeon had torn the skin off to let the red veins and capillaries quiver in the scalded air. Help, help! Fire! Run, run! Heat snapped mirrors like the first brittle winter ice. And the voices wailed. Fire, fire, run, run, like a tragic nursery rhyme, a dozen voices, high, low, like children dying in a forest, alone, alone. And the voices fading as the wires popped their sheathings like hot chestnuts. One, two, three, four, five voices died.