Forrest's simple speech about his mother's death. He doesn't go into too much detail, but the little hitch in his voice conveys just how affected he was and is still saddened by losing her. And the woman he's speaking to has to wipe away tears.
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.
Forrest: Why don't you love me, Jenny? I'm not a smart man, but I know what love is.
When she was a child, Jenny and Forrest runs into the corn field to hide from her dad. They pray.
Jenny: Dear God, make me a bird so that I could fly far. Far, far away from here.
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.
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.
According to Hanks, this was what Forrest said to the crowd.
Forrest: Sometimes when people go to Vietnam, they go home to their mommas without any legs. Sometimes they donít go home at all. Thatís a bad thing. That's all I have to say about that.
Lt. Dan at the wedding, walking on new legs, having found love and financial security, 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.
Even better, his fiancee is implied to be Vietnamese, mirroring real life attempts between American and Vietnamese veterans of the war to heal and make peace with one another.
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."
Seeing Lieutenant Dan broken and legless after Forrest saves him.
Lt. Dan: I was supposed to die on the field! With honor! It was my destiny, and you...cheated me out of it! I was Lieutenant...Dan Taylor...
Forrest's reply was pretty moving and poignant in it's simplicity.
Forrest: You're still Lieutenant Dan.
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.
Lt. Dan: "What am I gonna do now?"
Forrest's mother telling him to come home safely before he's sent to Vietnam. The way Forrest leans against her shows even he realizes he may not be returning home alive.
For any fan of The Beatles or his solo work, Forrest telling the story of his encounter with John Lennon is sure to come across as a sad moment, especially when Forrest recollects the circumstances of his murder years later. The tragedy and senselessness of his death is conveyed perfectly by Hanks' voiceover.
Forrest (narrating): Some years later, that nice young man from England was on his way home to see his little boy and was signing some autographs, and for no particular reason at all, somebody shot him.
And to add to the sadness of the moment, as Forrest narrates the line above, the camera focuses on Lennon, who seems to stare off in silent contemplation of something before the screen suddenly starts going staticy at the end of the narration, obscuring and eventually swallowing up Lennon's face to the viewer.