TL;DR : It was a combination of the two - Logan's healing factor + Jean's subconscious... but mostly Jean.
The scene in question, and in fact much of the plot for the movie, is loosely based on the original Dark Phoenix Saga that took place in The X-Men #129-138 waaaay back in 1980. At the end of that saga, Jean Grey
realized she couldn't control the Phoenix and committed suicide to stop it.
Within the context of the movie, however, it would seem that Jean was attempting to fight the Phoenix personality from within. Furthermore, she had previously asked Logan to kill her when she awoke in the Mansion:
JEAN: Where am I?
LOGAN: You're in the mansion. You need to tell me what happened to Scott. Jean, tell me > what happened to him.
JEAN: (remembering) Oh, God.
LOGAN: Look at me. Stay with me. Talk to me. (cups Jean's face) Look
at me. Look at me.
LOGAN: Focus. Focus, Jean.
JEAN: Kill me.
JEAN: Kill me before I kill someone else.
LOGAN: Don't say that.
JEAN: Please. (room shakes)
LOGAN: Stop it.
JEAN: (crying) Kill me.
LOGAN: Stop it. Look at me. Look at me. Jean. It'll be all right. We
can help you. Hey. The professor can help. He can fix it.
JEAN: (reverts to Phoenix personality) I don't WANT to fix it.
Also, at several places throughout the film it would seem that the Jean personality was influencing the Phoenix personality, just enough to rein her in. It should also be noted that the Phoenix personality appeared to show a preference for Logan over Scott.
When Logan was attempting to reach Jean at the end, the Phoenix personality (or perhaps the Jean one) seemed almost impressed that he kept coming despite her attempts to stop him. Given her raw emotional state, the Phoenix personality probably found his persistence... invigorating. By the time he reached her, her thoughts most likely resembled the cover of a romance novel.
So it's entirely likely that one of the personalities - if not BOTH - allowed Logan to reach her on purpose, albeit for different reasons.
FYI - In the comic storyline, the final battle played out differently. Long story short, while on the Moon Wolverine threw Colossus at Jean (a "fastball special") and Colossus's armored punch brought Jean to her senses long enough for her to commit suicide.