Appealing to someone's emotions is a powerful method of influence in the real world. So it can be in a roleplaying game too. At least potentially.
You do need to have players roll with it, though, as it's very easy from the meta-level of a player to simply refuse to be swayed. Particularly if the person doing the appealing is trying to convince them to do something they really don't want to do. As a GM, consider giving some sort of roleplaying award if a player has their character succumb to an emotional appeal, against what would normally be their better judgement.
Commentary by Keybounce (who has not seen the movie)
[Keybounce's comments will appear here when received.]
Commentary by memnarch (who has not seen the movie)
Finn used sympathy! It's not very effective! Maybe if Finn's explanation was more about taking the spy job to get away from the First Order and hunt for Gil's killer that could have turned out better. Claiming to be a friend might have worked if there had been more development of a friendship. Perhaps there's more of a developing friendship in the movie (that just fell apart in this scene) but up to now in-comic, it's looked like Rey has saved Finn's butt a couple times without Finn risking himself to save Rey in return. What kind of friendship is one-sided like that?
And wait, really? Finn's just leaving with the aliens? The ones who work for the First Order whose leader he thinks will kill him and something similar probably threatened in the movie as well? In spite of the obvious Pathos in the scene, something seems a Mite fishy here. Maybe Finn's going to ditch these two, head back to the Falcon, and lie to Chewbacca about everyone else being killed so they'll try to blow up the bar while leaving the planet! Sure the map will probably get lost, but killing some high ranking Resistance members might be a way to save his skin.
Transcript
Finn: {distraught} I lost my mate Gil.
Finn: I... lost my chance to bring his killer to justice. I thought I found a friend, but now I’m losing you too.
Rey: Your mind tricks won’t work on me. Go. I never want to see you again.
Finn: I slowly turn and leave.
{Finn leaves with Quiggold and Sidon Ithano}
Finn: Sorry guys, I—
Quiggold: Shut up. What happened?
Finn: You blew my cov—
Quiggold: Shut up. We’ve contacted Lord Ren. He’s gonna want a word with you.
{beat}
Quiggold: What’s the matter? Loth-cat got your tongue?