I think the actor argument is the best one there is. I mean, sure, every once in a while you're going to have your George Reeves, but for the most part, actors realize that they're just roles to play.
As far as trying to become more like our characters go, well, that's harder to argue, because one would hope that the game is educational as well as entertaining, and that by "becoming" other people we enrich ourselves by understanding their perspective. And some of that DOES become part of us, after a while - hopefully, the better parts.
Far from making us less mentally stable, RPGing may actually brace our minds to deal with situations that we have yet to have to deal with in real life. The very few of us that lose it in the game are probably mentally unstable in the first place. I KNOW I was:
My mother was and is a socio/psychopath. She is currently serving a term in prison for hiring someone to kill one of my older friends for money that he had made the mistake of making her the inheritor of (LONG story), and has no chance of parole until 2019 - by which time my kids will be grown and well able to defend themselves, thank goodness. When I was little, she used to take a lot of the little things that I would do - things that all little kids do, like sneaking their dog in the house, or drawing in crayon on the wall - and tell me that they had to be covered up quickly or my dad would beat her. This pattern went on until I was well into my teens. She would "produce evidence" of these things, too, such as being incredibly weird or moody to my dad until he would get angry and yell (he DID have a temper), and then late at night she would scream and bang herself into walls and such. She would use a syringe and put makeup into her skin under her eyes and tell people, including me, that he had beat her. And, as I said, any little thing I did wrong was "in danger of getting her killed, because she wouldn't let him hurt me." And I witnessed her commit many many crimes - shoplifting, price changing, giving little old diabetic ladies candy to get them to give her stuff while they were screwed up on sweets. And of course, all that fell under the category of things Dad couldn't know about, too, even though they were "his fault, because he wouldn't give her money to spend" and he would "beat her if she spent the money to buy me a soda."
So when I was about 13, I had started playing D&D again (I had played it when I was 10 briefly with the family of a friend - they played it as an educational family game) only this time my DM was very much like the woman in the Chick Tracts - abusive and commanding of his players, and thought to be involved in cult activities like the annual Halloween pet decapitations that were happening in town. (And this, for the benefit of your friend if you have her read this, is far from typical behaviour for a D&D gamer, but very typical of a psychopathic jerk.) Between this and all that was going on at home, and perhaps puberty, I snapped, and went through a period of about two weeks when I legitimately BELIEVED I was the Ranger I had been playing. And to be perfectly honest, I am at this point in my life uncertain whether I ever "came out of it", or whether I just relearned to "be me".
I do know that it was at this time that my parents stopped having such an easy time dealing with me - I was no longer tolerant of any of her illegal activities, and she began having to sneak around me (since she took me everywhere with her to try to keep me under her influence) to get any of it done. I was no longer putting up with Dad's occasional threats to kill himself - I asked him to please do so, because he was a dishonorable liar and I was tired of hearing it. (He and I get along much better since I found out everything she was up to, and he no longer threatens to kill himself - at least, not to me.)
So, as you can see, I "lost it" a bit, but I was mentally unstable to begin with - if anything, D&D gave my mind the tool it had been looking for to stabilize with. And now I'm a mentally healthy if somewhat eccentric father of two great kids who will NEVER have to deal with anything like I did....
