After a night of self destructive drinking a boy (Chris) discovers that his girlfriend (Anna) possesses the ability to remain immortal as long as she does not fall in love. As Chris’ family life decays, he runs away from home with Anna to show her a life bereft of love is not one lived.

as long as she doesn't fall in love....with him or someone else?

clear that up and you have a clear log line in my opinion!

I like the concept, but what I don't understand is why the two main characters are still together if one doesn't truly love the other?

I'm clear about what the characters are trying to do. However I'm not clear on the motivations or the point of contention.

There is a log line formula that I just used (dug it out after forgetting about it):

Here it is:

A (protagonist) has (a descriptive problem) and must (achieve goal) to solve (problem).

Where the words are parenthetical, fill in with your character and plot points.

Maybe that helps.