Chlorine gas (green dragon's breath weapon) is HIGHlY corrosive. The main way that it kills somebody is liquifying their lungs causing asphixiation, but it will still cause quite a bit of damage as it disolves soft mucus membranes like the inside of your nasal passage, the lining of the throat, and eye lining and retinal tissue on contact, and seriously burning skin tissue. To quote a haz-mat info sheet "Chlorine reacts with water and tissues to form corrosive hypochlorous acid and hydrochloric acid."
I would think that a creature holding his breath would still take the full damage from the gas, since you only really stop the gas from entering the lungs and it still will easily burn the soft tissue and skin, and holding your breath for any great length of time while undertaking physical exertion is going to be VERY hard. Your average fighter wearing his 30 lb breast plate and wielding an 8 lb longsword that just charged 120 feet to take a swing at a dragon is in no shape to just stop breathing, and might very well pass out if he tried.
A creature that does not need to breathe might automatically only take half damage from such at attack, at the DM's disrection. He would still get skin and retinal burns if he were caught in the blast.
What not breathing really helps out against are things like being trapped in quicksand, trapped in the mine shaft without air, stench attacks of undead that paralyze, being underwater, and so forth. But it stands to reason that a creature that is not breathing could not talk while being immune to these kinds of effects, since talking does require breathing. Also your typical air gensai would not be to keen on hanging out underwater, since it is his opposed element.