If it doesn't burn the house down, it will probably wreck your chimney and scare you half to death. Here's how it happens:
Begin with one dirty chimney-- let's say a fireplace not cleaned for a couple of years. The soot lining the chimney flue is saturated with flammable material called creosote. (More detail on creosote is available in any dictionary.) It doesn't take much to ignite creosote -- a newspaper fire sending flames up to the damper is enough.
Once kindled, the creosote burns with joyous abandon. In a matter of seconds the fire spreads up through the flue creating a draft that only helps things along. At this point your average chimney fire begins to roar and howl, sending the occupants of the house running for cover.
If you run, you'll be treated to a real fireworks show. As the creosote fire builds and intensifies, the heat melts the mortar between the brick. The red-hot mortar actually drips into the flue, only to be caught up in the tremendous updraft, and these little fireballs shoot out the chimney and onto the roof.
If you're lucky, the chimney will fall apart, destroying one end of the building-- if you're not, this miniature volcano can bring the whole house down.
However, if you stand your ground, a chimney fire can be controlled. Put out the fire in the fireplace with an extinguisher or sand. Cover the opening with a wet blanket, call the fire department, and hope for the the best. But it should be pointed out that having your chimney cleaned is a lot easier on the nerves than the sand and wet blanket method.