I play a 3rd-level Air Genasi Monk. My party will be fighting a Green Dragon soon.
Can a Green Dragon's Poison Breath affect me if I'm not breathing due to the Air Genasi's Unending Breath trait?
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The language of the ancient green dragon's Poison Breath is (MM, p. 284; emphasis mine):
The dragon exhales poisonous gas in a 90-foot cone. Each creature in that area must make a DC 22 Constitution saving throw, [...]
All it says is that there is a poisonous gas, and anyone in its cone is affected. If holding your breath were an option to save you from the breath weapon, it would provide that caveat.
For a counter-example, please note the language in the description of the dust of sneezing and choking (DMG, p. 166-167; emphasis mine):
When you use an action to throw a handful of the dust into the air, you and each creature that needs to breathe within 30 feet of you must succeed on a DC 15 Constitution saving throw or become unable to breathe, while sneezing uncontrollably.
The poison gas doesn't require you to breath. However, as a DM, feel free to grant the Air Genasi advantage.
Note that poison gas quite often can harm people who don't breath; mustard gas, for example, causes the skin to blister (and not from acid).
Contact poisons exist. This being a dragon... we can assume that poison is poisonous in every way.0
However, your lungs are much less well defended against stuff like poison (they are more permeable; human skin really is a layer of armor), so not having to breathe would be an advantage. Which is sort of what advantage is for; when the circumstances would be more to your advantage than normal, you give advantage on the check.
As a player, you can ask for advantage, but the DM is under no obligation to give it to you. Asking for immunity would be a bit much.