Some of the songs on The Times They Are a-Changin', especially "The Ballad of Hollis Brown", when you consider that that kind of thing can often happen in real life, to say nothing of the real-life tragedies which inspired "The Lonesome Death of Hattie Carroll", "Only a Pawn in Their Game", and the unreleased "Death of Emmett Till."
The best example is probably just about every song on Blood on the Tracks, which was written shortly before (not after) his divorce. It is commonly considered to be a defining example of break-up music.
"If You See Her, Say Hello" will crush you on it's own as well. It's such a beautiful, thoughtful song. After an album that's bitter and angry, to hear him sing a song that's more grateful and introspective just shows how complex of a person Dylan really is.
The original New York take of "You're a Big Girl Now" that was left off the album but later released on the Biograph box set. The musical setting is more spare and Dylan's vocal sounds much more heartbroken. When he sings "I hope that you can hear, hear me singing through these tears", he really does sound like he's about to cry.
"Sara" is, on its own, a well-written and melancholic love song. But realizing that it was one of the few songs that Bob Dylan wrote about his personal life — in an attempt to salvage his failing marriage — makes the emotion within it a lot more powerful. And then, you find out that it didn't work.
"Sweetheart Like You," because of the implication that due to Humans Sucking and Crapsack World, the genuinely good woman he's singing to won't be able to make it in the world.
Similar to "Sara" is "Ballad in Plain D," another of Dylan's few songs about his personal life. Dylan reflects on his failed relationship with Suze Rotolo, coming to terms with the fact that it was as much his fault as anyone else's that it didn't work. The simple arrangement of voice, guitar, and harmonica means the song is positively dripping with regret and remorse, and it can be devastating to listen to if you're in the right (wrong?) frame of mind.
"Goin' to Acapulco", especially in I'm Not There.
The song "I'm Not There" manages to be emotionally affecting even though it was only meant to be a rough demo with Word Salad Lyrics.
Perhaps it's just me, but this troper finds the song "Billy 1" heartbreaking, especially the verse : "The businessmen from Taos want you to go down/They’ve hired Pat Garrett to force a showdown/Billy, don’t it make ya feel so low-down / To be shot down by the man who was your friend?"
Also: "Forever Young", "Knocking on Heaven's Door", "It's All Over Now, Baby Blue", and "Sad-Eyed Lady of the Lowlands".
And don't forget "Not Dark Yet" from Time Out of Mind. Get your handkerchief out.
Or "Don't Think Twice, It's Alright"
"Shelter From the Storm" has a one-verse Dark Reprise with a formerly compassionate and giving lover turning cold and harsh due to the narrator's treatment of her.
"Girl from the North Country", particularly the earlier version. A song about a lost love, it cuts to the heart.
The last stanza of "Blowin' in the Wind":
How many times must a man look up Before he can see the sky? Yes, an' how many ears must one man have Before he can hear people cry? Yes, an' how many deaths will it take until he knows That too many people have died?