Visit most modern stables and you'll notice that there are extreme precautions being taken against fire. No good horseman smokes anywhere near the stable and any source of heat is guarded carefully.
The truth is that a lot of the stuff kept around horses is flammable. Hay can spontaneously combust (which is why most people have stopped using hay lofts), especially if it's damp or moldy. Grain dust can also catch fire, or even explode. Common forms of bedding - straw, wood shavings and paper - are also flammable.
So, if you need a place for a fire to start, the stables are often a good choice. Sadly, barn and stable fires often result in the death of animals - particularly horses, who are prone to panic.