Aeris dying in FFVII (I thought I'd be the only one, as everyone here seems to not care/joke about her death, if the Log Out picture is any indication), and the Eyes on Me seen in FFVIII. I am sure there is more.

I cried during the opera scene, and the suicide attempt scene while playing a fanmade version of FFVI based on a TV show, I can't remember if I cried at the real game. I doubt the suicide attempt would have made me cry in the real game because made to look like a fun acctivity, not a suicide attempt.