As Joseph Weissman stated in the comments, at the root of it is a false dichotomy. I'll look at this formally, but if you think about the sort of polarised spirit that is usually to be found behind statements such as "If you're not part of the solution, you're part of the problem", you can begin to see yourself.
An equivalent formulation to ¬S(u) ⇒ P(u) in classical logic (being a material implication) is S(u) v P(u): that is, either you are a part of the solution, or you are a part of the problem — leaving no room for anything which is neither (but, incidentally, leaving room for things which are both, which for societal problems might have examples in well-meaning but uncritical zealots). A dichotomy is exactly such a proposition which holds of all variables: in this case, ∀x: S(x) v P(x), which in the old saw is instantiated with the value x = u = you. If the universal statement is unsound, however, it is a false dichotomy.
In informal usage, a "false dichotomy" is still essentially the same as what I've said above, because the phrase 'false dichotomy' is only common currency among those who study logic, or rhetoric (e.g. in the form of law). How many problems are there, which are so clean-cut and simple that everyone is always either making the problem clearly better or worse, and never acting in such a way that either has no impact, or more perhaps having an impact whose value is quite ambiguous? Not to mention the often unrecognised complexity of assessing what the actual impact of someone's behaviour is.
Of course, statements such as "if you're not part of the solution, you're part of the problem" are not really intended to be formally sound. They're catchy phrases which are meant to stick in your head in order to make you more conscious of your behaviour — and the content of the idea precisely is that if you're not conscious of your behaviour, you're likely to be perpetuating some problematic system. So oversimplified statements such as this perhaps serve a positive role: they are part of the solution.
But the value of such stock phrases come from not evaluating the phrase itself very critically, when uncritical thinking is exactly what it is meant to combat. (The consequences of assuming that it must have formal, critical value is the basis for Wienersmith's joke.) If one assumes that such stock phrases represent critical thinking, one can become preoccupied with interpreting them as received texts, rather than doing original and free thinking about whatever subject may be at hand. Phrases as "if you're not part of the solution, you're part of the problem" can itself be part of the problem, if one is concerned with critical evaluation of one's behaviour. But if one already accepts that the impact of behaviour can be complicated and difficult to assess, the recognition that using a stock phrase has ambiguous utility shouldn't be too shocking.