There are far too many to count because I tend to easily attach myself to members of my team in video games, so their joys and pains become mine (yeah, it's odd... but it makes the games much more enjoyable) Here's the main ones I remember:
FFVII - twice. The first was from the pain of losing Aeris, of course. I was actually quite young at the time (9 years old), and had to stop playing because I depended on her healing so much and couldn't alter my strategies. Second was when I finally beat it and was just ecstatic!
FFIX - Pretty much the entire ending... from joy (and the music).
NIER - Yeah, if you've played you know that this can't really be narrowed down to any particular moment. Although, this was the first game to actually make me feel rage at the opposing team (from the sacrifice of that one character)
Kingdom Hearts series - This series tends to do this to me, though I fully understand the cheesiness of quite a few of the scenes.
Mother 3 as well - esp. the first three acts and the final act.
For some reason, I didn't tear up that much during either of the ME's. Ah well, they're both still great games!