No he does not get hurt while fisting the enemies.
The flames don't do immediate damage, only when the arch viles attack animation finishes. That long you have chance to hide around the corner. As long as the arch vile doesn't see you, it can't do any damage.

Come on guys, I thought you PLAYED Doom after all!?
Believe me, there are no cheats involved.

Except for a few monsters like the chaingun guys and the huge chaingun spider, all enemies have visible shots and you get your chance to strafe and escape without losing health.
Not many modern games are like this.
Most times there are more or less invisible shots and there's a medipack around every corner as it is intended to get hurt.