The idea of a biological system for emitting flame as a defense or attack mechanism is pretty incredible but, surprisingly, there's at least one real world example of something similar, although far less dramatic. The bombardier beetle has a defense mechanism for producing tiny explosions that blast a mixture of hot liquids in a would-be predator's face. These liquids can reach temperatures of 100 degrees C. The tiny explosions are created using a complex system for storing and eventually mixing hydroquinone and hydrogen peroxide with enzymes. However, even if we concede that a flame producing mechanism, though unlikely, is at least conceivable, there are still numerous problems with the way it's depicted.
First there's the milkshake problem. Any energy transfer out of the dragon in the form of flames must first go into the dragon in the form of food. If we assume an energy conversion efficiency of 100 %, the dragon will have to consume about 60 milkshakes in order to expel the energy contained in a gallon* (3.8 liters) of gasoline. (Yes, we know the dragon doesn't use gasoline but it's a convenient unit of measurement.)
The complex process of converting food to fire would probably not be 100 % efficient. In other words, the dragon will have to take in considerably more energy in the form of food than he expels in the form of fire. For our purposes we'll assume a 50 % efficiency of converting food energy to fire energy.
In one scene, a dragon torches an entire convoy of armored vehicles melting tires and killing almost everyone. This required an enormous outpouring of energy which easily had to exceed the energy of 100 gallons (380 liters) of gasoline. The dragon would have had to consume the food energy equivalent of 12,000 milkshakes to make this one flame throwing run. He'd have to own a chain of Dairy Queens and spend several days ahead of time slurping shakes.
There's also the flame fallacy to contend with. A flame does not have to impinge directly on an object to heat it up. A massive flame such as those expelled by the dragons would emit very large amounts of radiant heat in every direction including backwards. The dragon itself would get burned by the flame unless it was protected by a combination of heat-resisting insulation and a cooling system. It would not only need fire resistant scales but would probably also have to sweat massive amounts to keep itself from roasting.
Firefighters are well aware of this problem. The heavy bunker coats and other attire they wear are designed to insulate them from radiant heat. In addition, they will often wet down their coats before going near a particularly hot blaze, and depend on the water spray of their hoses to keep them from getting burned.
Propane gas flames are one of the hottest blazes fire fighters can face. These can happen at propane tank filling facilities when a flexible propane hose breaks. If the gas ignites, the end of the pipe becomes a giant blow torch. Firefighters are trained to walk up to this inferno by spraying a fine water mist in front of them with a fire hose. A firefighter then shuts off the nearest gas valve while his teammates keep the water spray going. Usually, additional hoses are trained on the fire to keep the area as cool as possible. If the water supply fails the team near the propane flame can receive burns before backing away even though they're wearing protective clothing and the flame is not impinging on them.
