EDIT: you guys have dug up some truly horrible pisstakes :D Thank you for those.
To the serious folk - relax a little. This is Mildly Infuriating, not I'm dying if this doesn't stop. As a non-native speaker I was taught a certain way to use the language. The rules were not written down by me, nor the teachers - it was done by the native folk. Peace!
The trouble with double negatives in I think Germanic languages in general is that they’re possibly ambiguous, relying on either tone and context or complex grammar to disambiguate whether you mean to negate a negative or mean to pile them up. Also negating negatives should be avoided if you can say things straight-up, there has to be plenty of reason to choose “Don’t not go there” over “Do go there”.
But that’s all style. It has also been said that you should describe how things are, not how they aren’t, and then Douglas Adams comes along and describes a space ship as “hanging in the air in the way that bricks don’t” which is pure brilliance (because it says, in negative space, something else about what that ship is: Eerie to the onlookers). Rules are there so you stop and think before you break them. If you want to write like Douglas Adams just make sure that you always wait until the traffic light turns yellow.
One important thing to realize is that different dialects of English have slightly different grammars.
One place where different dialects differ is around negation. Some dialects, like Appalachian English or West Texas English, exhibit ‘negative concord’, where parts of a sentence must agree in negation. For example, “Nobody ain’t doin’ nothing’ wrong”.
One of the most important thing to understanding a sentence is to figure out the dialect of its speaker. You’ll also notice that with sentences with ambiguous terminology like “he ate biscuits” - were they cookies, or something that looked like a scone? Rules are always contextual, based on the variety of the language being spoken.
How would Appalachian English say “Don’t not go there”? Even languages that only use negative concord have constructions to do double negatives, in Russian that’s done like “This is not unprovable” vs. “This is provable”, with “un-” (“без- / бес-”) being a very productive modifier. Sometimes the double negation becomes so common that it becomes part of the word, say небезопасный, “nonundangerous”.
I would expect, in practice, something like “Don’t stay away from there” but as we’re talking about a dialect continuum it doesn’t sound terribly unlikely for people to simply switch grammar (not necessarily phonetics or lexicon) to a dialect in which there’s no negative concord. And that, mostly, is what I mean by “ambiguous”.