I too can rather be sentimental, albeit more for sad occasions over happy ones. There's often times I'll need to avert my gaze when watching movies with friends, since I'd likely weep over what they couldn't imagine doing.
Shame on me, I guess.

As well, one could call me a tragedian of sorts; I enjoy these things in fiction because if something can get the eyes flowing -- or simply come close to it -- then I feel it does the job for having such a profound effect on me.
That being said, here's some prime candidates:
The Grey (2011)Here we have an early sequence where a team of oil drilling workers were flown enroute to their job site, only for their plane to crash in midst a frozen wasteland. While recuperating, they come across someone on their last throes:
https://www.youtube.com/watch?v=h2FB8f_J_U8There's a lot of trepidation here, but with all the things they do to ease him across, it's the
very question Liam Neeson's character asks is what kills me. This is easily one of the more intimate, grounded portrayals of death I've seen.
Saving Private Ryan (1998)Frankly, there's more than enough harrowing things in this to send stone-faced WW2 veterans packing for a life-time. None of our experiences could hope to compare, but for the sake of viewing, this death was truly hard to watch:
https://www.youtube.com/watch?v=NSRr7wUjLxwTwo men fighting with one knife on hand; it gets to a point you don't even care who you're fighting for anymore -- it's either life or death. What made it worse was this kind of mutual, yet twisted comfort the German lends his victim.
The Shawshank Redemption (1994)My favourite movie of all time, as it delves into the deepest ends of hope and despair. Here we're treated to the monologue of an elderly man released from prison, the setting in which this film takes place. It's real as it brutally gets:
https://www.youtube.com/watch?v=kotNxb2YApkThrough the decadence of life, finally comes silence. I give great applause to this actor -- nobody else could have sold that scene and made such a lasting impact as he did. Beyond anything else, though, this man does not die in vain.