Every single death scene. Plus, when Forrest meets his young son and he asks if the boy is "smart or is he..." and can't bring himself to finish.
Peter Scolari, Tom Hanks' co-star on Bosom Buddies, mentioned in an interview that this was the moment that did it for him; Forrest is at least smart enough to know how he is, and is smart enough to fear passing that trait on.
Jenny, after years spent being abused by her father, running away and experimenting with drugs and giving her body to men who didn't care about her... returns to Forrest's home in simple clothes and with a purse, politely greets him... and then runs desperately into his arms, hugging him close.
"You died on a Saturday morning..."
When Forrest leaves the letter at Jenny's grave. The pain and emotion in Tom Hanks' performance in the scene is absolutely amazing. He just sounds so broken, like he doesn't know how to keep going, but knows that he can't give up either.
Forrest & Jenny reunited at the Lincoln Memorial.
His speech right before is an in-universe example. We never get to hear it thanks to some conveniently-timed microphone sabotage but those around him, including Abby Hoffman, are moved to tears by what they hear.
Lt. Dan at the wedding, walking on new legs, finally at peace. Gary Sinise has since turned his entire character in a CMOH because of his work for the VA and USO, especially suicide prevention hotlines.
The first time Forrest meets Jenny on the bus to school.
Forrest (narrating): You know, it's funny what a young man recollects. 'Cause I don't remember being born. I, I don't recall what I got for my first Christmas and I don't know when I went on my first outdoor picnic. But, I do remember the first time I heard the sweetest voice in the wide world.
Jenny: You can sit here if you want.
Forrest (narrating): I had never seen anything so beautiful in my life. She was like an angel.
Close to the end of the movie when Forrest is seeing Lil' Forrest off to school. He tells Lil' Forrest he loves him and says he'll be sitting there when Lil' Forrest gets home... and he actually does.
Jenny's breakdown when she starts throwing rocks at her childhood home can also count, too. Even though Forrest has it bulldozed later, he never truly understood why she hated it as much as she did, but he did realize that she did not have good memories of the place.
Forrest (narrating): "Sometimes, I guess there just aren't enough rocks."
The death of Bubba, while now used as one of the biggest examples of Black Dude Dies First, was still pretty sad. Especially his famous last words.
Bubba: I wanna go home.
Lt. Dan, after pulling Forrest from his bed to the floor, angrily saying he was supposed to have been killed in the war instead of losing both legs and Forrest cheated him out of it, and sobbing against Forrest's chest.