There are no human characters in the story, and so in a literal sense there cannot be a "Man versus" conflict. However, humanity is seen in their achievements, shown here by an automated house that exists for the sole purpose of making life easier for humans. However, the humans are gone, so the house must simply work, moving along as it has been taught without any higher purpose. At the end of the story, the house tries in vain to fight a kitchen fire:
...there were twenty snakes whipping over the floor, killing the fire with a clear cold venom of green froth.
But the fire was clever. It had sent flame outside the house, up through the attic to the pumps there. An explosion! The attic brain which directed the pumps was shattered into bronze shrapnel on the beams.
(Bradbury, "There Will Come Soft Rains," nexuslearning.net)
This last-ditch effort to fight fire shows the human aversion to being burned and the lengths to which humanity will go to protect themselves from accidental fires. In a more symbolic sense, then, this shows Man versus Nature, as Man's creation (the house) fights against a natural accident (the branch which blows through a window and spills oven cleaner on the stove). Since there is no human brain to make decisions based on instinct instead of pre-programmed responses, the fire (nature) wins in the end, leaving Man (the house) destroyed and forgotten.
Nature vs. Technololgy
or Man vs. Technology