Here goes off the top of my head:
Rick Blaine lets Ilsa go. Casablanca (1942).
Fredo Corleone's last moments on the lake. The Godfather, Part II (1974). Really shows you how far Michael has fallen since he was an idealistic young soldier at his sister's wedding.
Chief Bromden smothers McMurphy and then makes his great escape. One Flew Over the Cuckoo's Nest (1975).
Oskar Schindler realizes he "could have done more" and we see how many Jewish lives a nice pen or gold-plated cigarette case were worth to Nazi officials. Schindler's List (1993).
Forrest at Jenny's grave. Forrest Gump (1994).
William Wallace seeing his wife in the crowd as he dies. Braveheart (1995).
Count Almasy's lethal dose of morphine. The English Patient (1996).
Jack dies. Titanic (1997). Despite having hated this movie and the fact that it only deserved its technical Academy Awards, this is a sad scene.
Joshua believes he has "won the tank." Life is Beautiful (1998). Really brings home how hard Guido worked to shield his son from the horrors of the Holocaust, and how successful he was despite his sacrifice at the end.
The letters scene/Mrs. Ryan is informed of her sons' deaths. Saving Private Ryan (1998). I still can't believe this movie lost out to Shakespeare in Love for Best Picture.
Maximus dies. Gladiator (2000). Or maybe this is a happy moment, since he finally gets to be with his family.
Shughart and Gordon are shot and their bodies paraded around by Aidid's militia, Durant get captured while reaching for a photo of his family. Black Hawk Down (2002). Didn't seem sad at first, until I read that this is actually how the real Sgts. Shughart and Gordon died. Also, Shughart is the one who calls his wife before the mission.
Padme dies. Star Wars - Episode III: Revenge of the Sith (2005). Star Wars has always been the story of Darth Vader, with each other main character representing a facet of his personality. Luke is his conscience, Obi-Wan his frustration at his own limitations, Palpatine his dark side, and Padme his goodness and humanity. With Padme's death, he becomes "more machine now than man."
And two that are supposed to be sad but are just depressing:
Mrs. Mulwray gets shot. Chinatown (1974).
Nick shoots himself in the head. The Deer Hunter (1978).
And finally a few that I can't remember very well, but I know have their moments:
Dr. Zhivago
Love Story
Hey - What about -
Spock's death (Star Trek II - The Wrath Of Khan.)
The destruction of the Enterprise (Star Trek III - The Search For Spock.)
In Dr. Zhivago, he drops dead while chasing Lara on a street. Lara never notices.
In Love Story, Jenny's death is off-camera, and it's right that when Oliver (Ryan O'Neal) says the famous line to his dad (Ray Milland) "Love is never having to say you're sorry."
Thanks for reminding me, Adam Bomb. I have to see those again some time.
A few more:
The Massachusetts 54th marching off to battle for the last time. Glory (1988).
Bambi's mother gets shot. Bambi (1941).