According to Riddle Spot, the answer to the riddle "What goes up the chimney down but cannot go down the chimney up?" is an umbrella. An umbrella is very thin and narrow when it is down, so it can fit through the narrow opening of a chimney. When an umbrella is up, however, it is too large to fit through the same narrow opening.
There are a couple of variations of this riddle. Some also include the idea of the umbrella going down the chimney down or up the chimney down but not being able to go down the chimney up or up the chimney up. The origins of this riddle are questionable, but it may be a reference to the storybook nanny character Mary Poppins, who went up the chimney with her umbrella.