Lonley Boy - *** Pistols. Love Will Tear Us Apart - Joy Division

I think MCR " I don't love you" is pretty sad.

it's heartbreaking to me that you would call that garbage punk

Too Drunk Too F#ck - The Dead Kennedys
If You Love Someone, Set Them On Fire - The Dead Milkmen

Also I have to agree 100% with MCM's post.

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Madina Lake - One Last Kiss (:

just to say
bayside is gross
and

"your own disaster"
by Taking Back Sunday

I guess it all depends on what you consider "heartbreaking." Also, what you would consider to be punk or rock. For example, you mentioned My Chemical Romance (my favorite band), and I personally think their MOST heartbreaking song is actually "Cancer." However, I don't know if you would consider it in the same category as most of their other music, because it's a ballad that's mainly comprised of piano and Gerard's vocals.

As for me, some of the more heartwrenching songs I listen to are:
"Blue and Yellow" by The Used
"Even in Death" by Evanescence
"Hemorrhage" by Fuel
"Breaking the Habit" by Linkin Park
"I Don't Love You" by MCR
"Helena" by MCR
"Forever" by Papa Roach
"It's Been A While" by Staind

Don't know if any of these fall under the genre you're looking for, though.

the cure-lovesong

There Is A Light And It Never Goes Out - The Smiths

cat and mouse - red jumpsuit apparatus