It did when I was younger. Mainly because I was afraid my dad would die, too, because he has HIV. Typing that just made me tear up, alas. Thank God for medical advancement.
I'm not a big Gump fan at all and I've always been puzzled by how beloved it is, BUT I do get a little verklempt when he asks if little Forrest is smart. That is pretty much the climax of the film for me.