Don’t @ me is an expression that originated on Twitter but is now used more broadly.
When you respond to someone or direct a comment to someone on Twitter, their @tag will appear in your tweet.
Don’t @ me, then, means “don’t respond directly to me” but more specifically it implies that the author is arrogantly not interested in hearing anyone’s response, so it is often used after dropping a controversial or bold statement.
It’s like ending a communication with “Out” or “end of!” or “no further correspondence will be entered into”.
“Language is fossil poetry don’t @ me” - Emerson
Don’t @ me has been around a few of years now on Twitter but I’ve been hearing it used in speech: in critic’s podcasts and so on. The @ is pronounced ‘at’. Apparently Jme had a song called “Don’t @ me” in 2015 but I was unaware of that at the time.
More recently I’ve been hearing the verb @ being used in other phrases, e.g. in a recent fivethirtyeight podcast I heard “feel free to @ me”.