Well, I was talking about GTA, but...
Now, I've never seen the Warriors or played the game, so I won't say anything about that.
But in Manhunt, James Earl Cash does have a heart, I would say. I mean, not exactly a "goodie goodie" kind of heart, but a heart full of emotions. He has passion in his work (notice how well he pulls off stealth kills), a sheer hatred for the director (not exactly a good emotion, but still), and doing all of these things so he can get out of the director's cruel game.
Saying that someone doesn't have a heart means that they're purely emotionless, kind of like a robot.