I don't think that death is the worst thing that can happen to a character.. Usually when a fictional death is sad to me, it's because of how it affects other characters
Even when I was little, I remember people talking about how sad the death of Bambi's mother was, and I totally gave no shits.
Oh, but now that I think about it, Mufasa's death in The Lion King was a big deal to me.
;-; the circumstances of Seymour's "death" in Futurama... (super spoilers: I'm happy they fixed it )
The first like... 10 minutes of Pixar's "Up."
The "death" of the 10th doctor (being reborn as the 11th). Wasn't so much "destroyed me," as I was just sad that he didn't want to go.
The kids in Grave of the Fireflies... I wasn't even sad, I just hated life for weeks after that stupid movie
Life is Beautiful (not going to say more because people REALLY need to see this movie)
But when a death is perfect to the story, even when I love the characters or am emotionally attached to the characters, I like to see them die (Kaori Yuki's Boys Next door is a good example)
More sad deaths : The Wolf in Dances with Wolves [;-;]
All the Stark dire wolves in A Song of Ice and Fire / Game of Thrones
Sam in I Am Legend (Film. Have not read the book).
(of all of these, I only remember crying for Up and Life is Beautiful. I think both instances were due to outside circumstances too. I might have cried at The Lion King, but I don't remember. Probably have.)
The Plague Dogs. I DON'T EVEN REMEMBER IF ANYONE DIES! It was just a depressing fucking book and made me hate life for the entirety of 3rd or 4th grade or whenever I was forced to read it!
Flowers for Algernon. Also, just depressing more than sad. :/