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.