What is the origin and meaning of the phrase can you not? To my ear, it has an archaic tone, but searches yield entries in the urban dictionary, along with one quote from Sense and Sensibility. Its twisted syntax, potential ambiguity, and indirectness intrigue me. Part of my language heritage is from Irish American, so I also wonder if this hints at its source.
I don't see why OP should think we need to find an 'origin' for such a standard construction.
Increasingly, people avoid using a negating not at the end of a sentence, as shown by this NGram, but I wouldn't go so far as to call the usage 'archaic'. A little 'dated', or 'formal', perhaps, since the modern style is generally more informal. People are more likely to say/write "Can't you" today, but it really is just a matter of style.
In addition to FumbleFingers' answer, there's the more sarcastic meaning of "can you not": that is, "please don't", as in, "Can you not play the piano so loud in the middle of the night?". In fact, this is how I parsed your question until I read the other answer (needless to say, I was surprised to apparently learn that Sense and Sensibility had such a hip style). I don't know how this version came to be, except through the usual method of sarcastically twisting a very weak statement, literally meaning "Do you potentially have the capability not to...", into a strong one.
In the documentary Terms And Conditions May Apply, Mark Zuckerberg asks the documenters if they are recording him, they say yes, and mark replies with "Could you not?". I believe the use of this word is ironically a subtle protest against personal information being given to big-brother and sold to marketers by internet bohemeths such as Facebook.