I have a plastic bottle that is impermeable to ${\rm CO}_2$ but slightly permeable to $\rm O_2$.
If I fill up the bottle with pure $\rm CO_2$ at $3$ atmospheres and expose it to $\rm O_2$ at $1$ atmosphere, will $\rm O_2$ enter the bottle? My understanding is that gases move from high to low partial pressure independent of other gases. Is this true?
In an ideal case in which particles don't interact, or in the limit of dilute gas, yes. Because the bottle contains no $O_2$ then the gas is more likely to enter the bottle than to leave it (because... there is no $O_2$ inside) so you have a net diffusive flow inside the bottle. That is, until the bottle reaches a (partial) pressure so high that the outgoing flow of $O_2$ (the molecules that was outside, randomly entered the bottle, and now randomly leaves) balances the incoming one.
In a more complex but more general case, where particles interact (either same-gas particles interact with each other or $O_2$ interacting with $CO_2$ (not always the case for these two gases, but for two other gases it could happen) and/or gas particles occupy a volume, then higher order effects might play a role (in the limit, if the bottle has no free space because the $CO_2$ is filling it completely then of course $O_2$ won't enter. But if you put in the numbers you see that for that to happen you need huge amounts of gas). Imagine you have 100 moles of $CO_2$ in that bottle. Then, if you approximate a $CO_2$ molecule as a sphere of radius $0.15 nm$ then you get that 100 moles completely fill out a 1L bottle, there is literally no "empty" space left. In that case, $O_2$ would not enter - but this is far from ideal/dilute conditions (and would also need a bottle with very strong pressure resistance)!
As a first approximation, you can use a Van der Waals gas model, where particles have an interaction and a volume, and then when the gas is not in dilute conditions anymore, the partial pressure concept fails.
But in the ideal case / dilute case, partial pressure law holds and $O_2$ would diffuse inside the bottle.
This is the gaseaous analogue of osmotic pressure with the plastic bottle playing the role of the semipermeable membrane. In equilibrium the ${\rm O}_2$ pressure will be the same within as without, and inside there will be ${\rm CO}_2$ pressure as well. The total pressure inside will therefore be be higher and may even burst the bottle.
