After a workout...What does it mean if a mans nipples burn?

Friction. Friction from clothing or sports equipment is the most likely cause of nipple irritation after active sports. It is easy to test. . . Just put a bandaid over one nipple next time, and see what happens!
Working out. Most probably, your shirt is rubbing on your nipples, creating friction and causing some local pain. Try wearing a shirt that is larger and looser fitting. If the problem persists, go to your primary care doc.
Is it your shirt? If you wear a shirt and are engaging in physical activity that causes mechanical friction against your breasts (i.e. The shirt is rubbing against your nipples while you exercise), then this may be the cause of your irritation. Most women wear a sports bra, so they don't have this problem as much. In some patients, if they are not used to exercising, sweating itself causes itching/burning.