Is there a rule about double negations that aren't meant as double negations (e.g. "We don't need no education")?

Doubled negatives are often used casually in certain dialects to indicate negative concord, an intensification of negation rather than an inversion of it. This typically happens when both words involved are simple negatives, and is most common with no standing in for a, an, or any alongside don't or ain't.

So you can safely assume that

He isn't not going to the concert.

is double negation proper, because it has emphasis, as is

She wasn't unimpressed.

because this is litotes, whereas

I ain't no hillbilly.

is negative concord, because it's obviously casual, and uses ain't no in place of am no or am not a. (It's also a patent lie, but that's beside the point.)

It’s worth noting that this “inverts” the sense only in Standard Written English. In AAVE and other vernaculars this is not only grammatically correct, it’s also semantically correct, i.e. the normal negation is in some contexts always created via double negation which does not resolve to a positive.

The same is true for many other languages: if you’re not convinced that a vernacular is a valid grammatical description (but from a linguistic point of view, it is!), take French, where double negative is mandatory (most of the time) by prescribed grammar rules: to negate a verb, it is embedded into “ne … pas” (or archaically “ne … point”) which are affix and suffix and both indicate negation, even when used in isolation (which does happen occasionally).

(Since this has garnered so many upvotes, let me give credit where credit is due, and also a reading suggestion: This information is from Steven Pinker’s excellent book The Language Instinct.)

He made this "mistake" of double negation on purpose. It is a form of protest in itself; what he meant was: we don't need education, we will speak (or do) as we want. He was going against the "imposed" correctness and etiquette.

What is correct may not be easy to understand, and what is easy to understand may not be correct. To a limit.

On reflection, it seems common that only no and ain't can be used to intensify a negation rather than a double negation, whereas correct English requires using any, isn't, or aren't.

I would probably never hear "This paper ain't unacceptable," but it's conceivable to hear "This paper ain't gonna get no acceptation from me."

Also, "This paper ain't unacceptable" is ambiguous, so I would expect to hear "This paper ain't unacceptable, but you gotta be proofreadin' some mo'."

Perhaps you can use exercises like asking what a certain written sentence might mean:

1. You're not wrong.
2. You ain't wrong. (still double negation)
3. I ain't gonna take any shit from you. (possibly ambiguous, a sign of someone trying to copy a dialect, but slipping up with "any")
4. I'ma takin' no crap from you.
5. I'ma takin' no attitude from none of youse.
6. None of you ain't right. (ambiguous)
7. None of you ain't wrong. (ambiguous triple negation)
8. Not one of you isn't wrong. (crystal clear triple negation)

The comprehension of an ambiguous phrase often depends on the context in which it appears.