A says to B "we should recycle because it is better for the environment," and person B says "if we wanted to be better environmentally, there's lots of things we could do that we don't, so we shouldn't recycle."
Is this a specific fallacy?
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The form of the reasoning here is: "we can not do everything, therefore we should not do anything", or more colloquially "nobody's perfect, so why bother". The commonly used names are the perfectionist fallacy and the perfect solution or nirvana fallacy:
"By creating a false dichotomy that presents one option which is obviously advantageous—while at the same time being completely implausible—a person using the nirvana fallacy can attack any opposing idea because it is imperfect. Under this fallacy, the choice is not between real world solutions; it is, rather, a choice between one realistic achievable possibility and another unrealistic solution that could in some way be "better"."
In addition to misguided perfectionism, there is another issue here, which is also potentially fallacious. "We can not do everything" is a more or less uncontroversial statement of fact, but "we should not do anything" is a norm, prescription. Deriving norms from facts, or prescriptions from descriptions, or "ought" from "is", in Hume's classical example, is generally a fallacy, called the naturalistic fallacy or Hume's guillotine. How things are need not imply anything about how they should be, or what we should do.
Very similar in spirit is the fallacy of relative privation, see What fallacy dismisses problems by presenting "bigger" problems? The perfectionist fallacy also has a curious opposite, known as the politician's syllogism:"We must do something. This is something. Therefore, we must do this". Clearly, neither doing everything nor doing something random is the right approach.
So Person B is likely at fault of; Moving the goalposts (raising the bar) – argument in which evidence presented in response to a specific claim is dismissed and some other (often greater) evidence is demanded.
This is because they dismiss that recycling will have enough positive environmental impact, they want evidence that recycling is better or as good as all the other things that are not being done.
Person B is by all means logically fallacious. How does our subjects' inability to do the "other" things to better the environment a preclude condition for this one thing they CAN do in which case here, it's the recycling?
Put structurally, B's premise is such that there a countless things people can do to better the environment which is undeniably true but doing the countless things is practically impossible. So he concluded that therefore we shouldn't recycle.
This is clearly non sequitur. There is absolutely no way the conclusion follows from the premise therefore a fallacy.