It looks like the US people locked themselves in the vault in order to destroy the documents. After they were done, they left. There might not be the oxygen in the vault to burn all of the documents. But that's conjectural.
According to http://en.wikipedia.org/Burn_bag , burning isn't so common due to environmental concerns and the high value of recycled paper, though some places have burn bags with built-in thermite charges, in case of emergency.
It's a good addition to any suite of destruction, but creating any kind of fire is a higher amount of risk for collateral damage than the other methods. Plus, unless the combustion is thorough, you may still be able to discern information off the paper.
