What? Any sensible interpretation of RAI and probably RAW is that a poison is only used as one vector at a time. But if you need any other justification...
Gas Mask protects against all Inhalation Poison
Tear Gas is Inhalation AND Contact
Therefore, Gas Mask protects against Tear Gas.
Think of it in MtG terms.
Card X deals 2 damage against all Goblins
Bob is a Goblin AND a Vampire
Therefore, Bob receives 2 damage.
Unless there's an "or" floating around somewhere I missed (possible) Tear Gas has the "Inhalation" tag. It doesn't matter what other tags it may or may not have--the Gas Mask doesn't ask about them. If there is an or that I missed, unless it is worded very carefully to the opposite, i'd say that it also supports a poison being only one thing at a time. I don't think there's any precedence for poison cycling through its various tags until it finds which one you're most vulnerable to. Now sure, does this produce some oddities? Yes. About as many as the idea that a poison is all of its tags at once and the gas mask does nothing.
I think the logic is a bit flawed though. If a toxin is attacking you via two or more vectors you do not get to pick which one it is going to use, circumstances will dictate which are in play.
Your wearing your gas mask and normal street clothes when you walk into a cloud of Tear Gas, the mask protects your lungs and face from the burning, but if there is a contact vector the rest of the body is exposed as the gas just passes through most regular clothing.
Some vectors it can be safe to assume they are not being shared. If a toxin is injection and inhalation, then a gas mask would protect from the inhalation and unless you have some kind of open cut injection would not be an issue (and even then it would be pretty thin). Contact is a bit more encompassing though and can fit well with other vectors like inhalation.
Now granted TG in real life is not that much of an irritant on the skin, but again this may be a souped up version so we will play as that for now.
In MtG terms
Card Yin/Yang deals 2 White Damage (Inhalation) and 2 Black Damage (Contact) but Damage doesn't Stack.
Bob is immune to White Damage (Gas Mask)
Therefore, Bob still receives 2 Damage from Black Damage (Contact).