I think this is dependent on context. If it's obvious to the recipient exactly what needs to be done and how, I can see "do the needful" being respectful and ha

I think this is dependent on context. If it's obvious to the recipient exactly what needs to be done and how, I can see "do the needful" being respectful and having the connotation of, "I'm not going to tell you how to do your job". Conversely, if you receive a vague or unactionable complaint followed by "do the needful", it has more of the connotation that you describe.