It's a negative imperative: it has a weird conjugation, and in the second person singular it uses the infinitive (informal as in here) or the subjunctive (formal, "Non me lo dica!"). Yeah, it's more a "you don't say!" exclamation, to stop someone from telling you something you'd have to add a direct object as in "Non me lo dire".
According to wordreference.com, it's an ironical reply like "You don't say" or "No kidding." here's the link
No, in that case you should say "non me lo dire/non dirmelo". "non mi dire", often precedeed by "ma", is an idiomatic expression that expresses something like "I can't believe it!" (= non ci credo/non posso crederci!). It can also be used sarcastically.
"Sua moglie lo ha lasciato? Ma non mi dire!"
"His wife left him? I can't believe it!"
You must append the clitics to all forms of "proper" imperative, not just "tu": however, the borrowed forms follow their own conjugation. The subjunctive doesn't append clitics, so the formal imperative (Lei) is "mi dica"; the infinitive allows both constructions, so it can be either "non dirmi" and "non mi dire". This particular sentence however is only missing an object because of idiomatic usage, if you break the idiom you have to add one, i.e. "non dirmi che..." or "non dirmelo".