I have no real answer for this, because I don't recall the movie clearing that up. But to me it looked like this.
Phillip thinks himself unworthy and only asks for her forgiveness, not for life, because he is that kind of humble. Unlike Blackbeard who wanted to sacrifice his daughter for his own life.
Syrena however tells him again to just ask. In my opinion, it's no really clear if she means for him to ask for forgiveness or if she refers to her earlier statement, that he only need to ask to be saved. He then asks for forgiveness bevor she takes him with her.
I thought this meant that she wanted to do more than simply forgive him. That she wanted to save him and be with him.
In some stories mermaids can grant the ability to breathe and thus live under water too.
So I thought she took him into the sea to be with him happily ever after. Because she did not seem sad when she took him down.
In this he would be a counterpart to Blackbeard, finding happiness in the end because he was selfless. Again unlike Blackbeard, who died beeing selfish.
Also, why would she take him with her, if she wanted to let him die? She could have just left him there. Unless for some reason she wanted him to have a watery grave. But I don't see why she would want that.