Punishing Fire's triggered ability triggers only if it's already in your graveyard at the time an opponent gains life. For example, you can cast it in response to a spell or ability that will cause an opponent to gain life. In that case, Punishing Fire will resolve first, so it will be in the graveyard by the time the opponent gains life. However, if you wait until an opponent actually gains life and then cast Punishing Fire, you won't be able to return it to your hand at that time.

You choose whether to pay {R} as the triggered ability resolves.

If Punishing Fire has left your graveyard by the time its triggered ability resolves, you may still pay {R}, but you won't return it to your hand. This is true even if Punishing Fire has left your graveyard and returned to it by the time its triggered ability resolves.