"My Body Is A Cage" has always been a tearjerker. It's even more of a triumphant one when Peter Gabriel performs it.
Off their first EP, "Vampire/Forest Fire". At approximately 5 minutes in, it just... takes off.
Funeral is legendary for this. Quite a lot of the band's popularity can probably be owed to the sheer cathartic power of their music (relative to the aloof detachment of most modern indie rock) so they have many, many more songs that could count as this. Specific examples include:
"Crown of Love" and "Wake Up". A one-two punch in the album order. The tone of the two songs is so different (bleak and uplifting, respectively) that one would think it would end up as Mood Whiplash... but somehow it makes perfect sense.
"Neighborhood #1 (Tunnels)."
"In the Backseat," which manages to sum up everything great and horrible about growing up in six minutes.
"Neighborhood #4 (7 Kettles)" definitely has this approach, too.
The later albums have their moments, including (but definitely not limited to) "Windowsill," "Suburban War," "Half Light I," and "Half Light II."
"The Suburbs (continued)" deserves special mention. Since it forms a bookend with the first track on the album, it calls back the original lyrics of "moving past the feeling". One has the impression that despite all the musings on friendship, growing up, and life, you are in the end very much alone. It's powerful stuff.
The fanbase knows "Ocean Of Noise" as one of the band's biggest tearjerkers. It's not just the lyrics, it's beautiful instrumentation.
"Sprawl I" can be both moving and unsettling. Arcade Fire has recorded many quite dark songs, but none (aside from perhaps "Crown of Love") are as bleak and uncharacteristically hopeless. The final lines can give you chills:
"Do you kids know what time it is?"
Well I said, "it's the first time I feel like something is mine
like I have something to give."
The last defender of the Sprawl said, "Where do you kids live?"
Well I said, "if you only knew
what the answer's worth
I've been searching
of the earth."
"Afterlife." It's about the desperation of trying to hold together a failing relationship. The dance elements of the song only make it sadder and more bittersweet.
"Here Comes the Night Time Part II", especially consider how substantial the mood whiplash from part I is.