What reaction is happening when a lithium polymer battery "is on fire"
I'll take a stab at this one, but I'm not a battery expert.

What is a lithium "polymer" battery?

Wikipedia says that "lithium polymer battery" is now commonly used to refer to nearly any lithium ion battery that is encased in a flexible polymer shell instead of a hard metal casing.  There is a chance for confusion here because it apparently used to mean a battery with a polymeric electrolyte instead of a purely small-molecule liquid-phase electrolyte.  I'm assuming you mean the former "in a polymer shell" definition for the purposes of my answer.

What type of anode, cathode, and electrolyte materials?

There are many types of lithium ion batteries.  Electrolyte composition, cathode materials, and anode materials can all vary.  Perhaps the most common is a lithium cobalt oxide cathode with a intercalated graphite anode.  Organic solvents must be used as electrolyte because water is not stable to the high voltages that lithium ion cells provide.  Solvents stable over a wider potential range must be used instead to dissolve salts that can carry electric current.

What happens when it's crushed:

So for lithium cobalt oxide batteries with liquid electrolyte, I was able to find a good source describing the problem of "thermal runaway":


  The temperature of the cell rises due to the
  chemical reactions between the organic solvent and electrode materials leading to cell failure/explosion. The chemical reaction is prompted because of the coexistence of 1)
  combustible organic solvent with inorganic salt electrolyte 2) lithiated graphite anode and 3)
  partially delithiated LixCoO2 cathode in the charged state. The graphite anode intercalate
  fairly well during the initial cycles. As the cycle proceeds, the available intercalating sites in
  the graphite slowly decreases and the lithium deposition over the surface of the anode
  increases during charging process. Such a situation prompts non-even lithium deposition or
  even dendrite growth over the graphite surface. When the battery is in the charged state the
  cathode remains in a delithiated state. Flow of current over and above the
  tolerable/standard charged state causes instability to the cathode which starts to release
  oxygen into the electrolyte. Thus released oxygen reacts exothermally with the lithium
  plated over the graphite anode and increases the temperature of the cell making the cathode
  to release oxygen further. The cell which was in the dormant state in the absence of oxygen
  and heat now becomes an explosive device in the event the cell is met with any abuse. 


That describes what happens when the battery is overcharged and leaks oxygen into the electrolyte.  I suspect that when the batteries are crushed, even if they aren't overcharged, that at least two things can happen leading to similar thermal runaway: First, the lithiated graphite is air- and water- reactive, heating the now-crushed battery and eventually lighting the electrolyte solvent on fire.  Second, the cell may be shorted by the mechanical disruption, so that even without air or water contact, the voltage in the battery can discharge very quickly.  This would create large heating from ohmic resistance, also eventually igniting the elctrolyte.