Metallica/Tear Jerker

Even this thrash metal band is capable of doing tear-triggering songs.

"Orion" by Metallica can be a tearjerker. It's a cool song, but what makes it a tearjerker is the fact that this is the song that was played at Cliff Burton's funeral after he was killed in a bus accident in 1986. Until recently (2006), the remaining members of the band had never played it live, partially as a tribute to their bandmate.
- Or, perhaps, because Jason Newsted simply couldn't play it well enough.
"To Live Is To Die" is another one. What makes this song special is that the band didn't sully their feelings by putting them in words. In the entire 10 minute song, the only lyrics are the following poem written by Cliff shortly before his death:

"When a man lies, he murders a part of the world.
These are the pale deaths, which men miscall their lives.
All this I cannot bare to witness any longer.
Can not the Kingdom of Salvation take me home?"

- The first ever live performance of "To Live is to Die" is intensely depressing.
"One": about a war solder who loses his limbs and senses, and has nothing to live for. Especially with the video.
"Fade to Black": about a man contemplating suicide, and follows through on it.
The straightforward ballad, "Nothing Else Matters".
"The Unforgiven" trilogy.
The video for "The Day that Never Comes" can be intensely depressing.
The guitar solo halfway through "Master of Puppets".
The music video for "Turn The Page" is another major downer.
"The God That Failed" becomes a Tear-Jerker when you hear the story behind it: it's James Hetfield's personal Rage Against the Heavens, his feelings after the death of his mother, who could have been saved if her Christian Scientist beliefs didn't forbid her from seeking medical treatment.