If you say "don't be mean" to someone that means, don't be nasty or unkind or rude. For example, If someone said to me," I think X is ugly and he can't dance". I would say "Don't be mean!" As someone pointed out it can also mean to be tightfisted with money, but most commonly it means unkind.

Mean as an adjective means a person who does not like to give things or to spend money ,it is opposite generous,so don't be mean ,means don't be like that person

I think it means don't be hateful.

Do not be rude (hurtfull)