It means that when things go wrong or he/she becomes upset, they don't cry about it.

It means that the person doesn't cry easily (or refuses to cry), or that they do not handle the sadness or tears of others well. Example for the first meaning: Even when John's dog died, he wasn't upset. He doesn't do tears. Example for the second meaning: Please don't cry. I don't know how to comfort people. I don't do tears.