I have a source saying positive commands usually occur in the perfective aspect while negative commands usually occur in the imperfective aspect - which is valid here :D.
"Don't talk to him about this"
"Don't say anything about this to him"?
Yes, should work.
"Don't speak to him about this"
Is not 'Don't talk to him' the same as 'Do not tell him'. My translation of 'Don't talk to him' was rejected.
Well, it's pretty far... in this sentence you can talk to him, you just should not tell him 'this' (whatever 'this' is).
Can it also be Nie powiedzcie mu o tym?
Not really, that sounds very strange to me. I'm not sure why, but I think that if you use imperative to say "don't do this", it's like "do not do this at all", not "do not do this at some specific occasion". So in that way it's imperfective.
Don't speak to him about it