You should shoot him if you're a long way from him and he's chasing someone else. Shooting him slows him down, to help a green survivor escape. But yeah I agree DBD, annoying when people stay and shoot (like bots), and get killed.
It used to be that setting him on fire doesn't do damage, so shooting him doesn't kill him faster unless you do 8000 damage, which you very rarely see on the "most damage to tank" page. He just dies after 40 seconds on expert when he's on fire, whether you did 0 damage or 7999. I don't know if they made him move faster on fire in the last update, I didn't see anything in the update notes.
edit :Quite a convincing video.. I guess that explains why I've always said the tank can outrun healthy survivors and other people say he can't. I was seeing him on fire always.