Supposedly Edward Teller was worried about the atom bomb setting the atmosphere on fire. Apparently he was wrong, but I was wondering what it would take to burn the earths atmosphere?
Scientists weren't worried about igniting the atmosphere, they were worried about the nitrogen and oxygen being sent into a chain fusion reaction. turns out even the center of the sun is not hot or pressurized enough to cause hydrogen and oxygen to begin fusion. Maybe the center of very old very large stars reach those kind of pressures and temperatures required to cause a fusion event with those elements. When those elements are fused it takes more energy to force them to do that than they put out when it happens too.
As for actual ignition, There is very little fuel in the atmosphere (nitrogen is not combustible, oxygen is not a fuel but an oxidizer, and other possibly flammable gases are in only minute quantities.) The atmosphere is also too efficient at diffusing heat for any sort of sustained chain combustion reaction to occur. To "ignite the atmosphere" you would need to heat it up uniformly across the entire (or at least most of ) the atmosphere at once. A nearby super nova might be able to do that, but that is going to obliterate the entire planet anyways so its a bit of a moot point.
Atmosphere as it is (i.e. without adding combustible) can be "set on fire" using three mechanisms:
the one Konopinski, Marvin and Teller worried about, a runaway fusion reaction involving either nitrogen, oxygen, or both. The conditions for such a reaction are too energetic to set up even with a fusion device, and if they were, those same conditions would also vaporize the planet.
a reaction between atmospheric N2 and O2, which combine to give e.g. NO2, thus asphyxiating every oxygen-breathing organism. This is a plot device in one of John Campbell's Aarn Munro novels (the Teffels try burning the atmosphere of their enemies' planet), but it will not work because the enthalpy of the reaction product is higher than that of the reactants by about 34 kJ/mol. You need to supply heat to make it work (that's why combustion engines, wherein there's heat aplenty, give off nitrous oxides, while a passive catalyzer is able to break them off again to nitrogen and oxygen without needing any energy).
a reaction between any suitable atmospheric component (nitrogen and oxygen being the only reasonable candidates) and another minority component, either naturally present or (semi-)artificially added, to yield a stable compound with negative net enthalpy change. This is posited in a ST:TOS novel, where the reaction is between oxygen and trace silica dust in the air. The book handwaves away the fact that the reaction is not thermodinamically possible (silica dust is already an oxide).
If you added combustibles enough, of course, it would work. That is 'what it would take'. But you would need a lot. About 25g of hydrogen for every kilogram of atmosphere (or 32g for each cubic meter of atmosphere) to get all oxygen burned to water. Between the heat and the drop in oxygen partial pressure, probably a tenth of that would make the Earth uninhabitable.