Hopefully without getting into semantecs on it, the short answer is TC , yes, iron, Fe, burns when exposed to Oxygen, it is Oxidizing, like rusting and it is also a chemical reaction.
rusting does not give off enough heat for you to notice that part of the reaction, but it is there. technically hydrated oxide of iron ( Fe2O2 +(HaO))
You can have fire underwater if using a product that produces or has its own embedded oxidizers, welding underwater for instance, OR a magnesium sparkler lit in air continues to burn under water, depending on the content, reacting and burning its own O2.
Without the Oxygen there is no Oxidation, you won't get a flintlock to go go off underwater simply because there is no O2 for the Fe to react with and burn/ oxidize rapidly, ie spark.
I am not certain of firesteel composition but I would expect part of the reaction to be Fe2+O2+ heat ( the strike) -->FE2O2 + heat ( or iron and heat and O2 combines to make ball of rust and heat in form of "Man that fire feels good!"
If I had known in school how much the chemistry applies in making and building muzzeloaders I would have been more interested!