@Veribal Dump? Dump. Mind you, my definition of heartbreak might be different to yours, but here are a few.
*A lot, and I mean A LOT, of these start slow. If you're picking any up, bear with them 'kay?
Heartbreak in small oneshot doses:
Replay.
Zansho; a few of the oneshot collection are heartbreaking but you could read the others too.
+1 to
Kodomo no Kamisama.
Ongoing stuff:
Gorae Byul, a historical manhwa about Japan-occupied Korea (Joseon) which uses symbolism from the little mermaid's story for an additional layer of depth. Stunning, but also painful.
Yuki ni Tsubasa; I do not usually recommend this, as it's really heavy, chock-full of monologue, and said monologue comes from a depressed kid. I really like the series though, so there you go.
Break Blade, a lovely manga with lots of characterization and progression for our young-adult MC when he is thrown into war.
Sekai no Hate is more along the lines of "scans are ongoing". It breaks my heart because you can see the bad end coming from miles away, even when our MCs are just doing their best to be happy. Poor kiddos.
Kiss & Never Cry, also awaiting scanlation, has really complex characters which you get easily attched to (...then you have to deal with all the hurt). Ogawa Yayoi at her finest.
Completed and all ready to break your heart (ordered by length):
49; an
extremely short read, almost like a oneshot. So you shouldn't expect too much.
Eien no With; same as above, but this one is more heartbreaking.
Magi ni Okurimono; about a guy who wants to remember his dead girlfriend and the girlfriend that wants him to forget about her. I WANT to say more but I'd spoil it, so just go ahead please. It's a personal favorite of mine.
Solanin; plain ol' life.
Familyman; a really sad story about familial bonds and a really cool normal dad. The official english is linked and free so grab that!
Good Night World, a really well-done series about the line between virtual reality and reality, that'll make you feel sentimental.
Koi-iji; the pain of unrequited love. I haven't actually finished this series (long overdue really), but going by the whole thing I can see how it will turn out. So yes, heartbreak it is.
Darker than the other completed ones (ordered by levels of darkness(?) ):
Piece, a manga about a university student who finds out that her supposedly plain classmate died, and that... she wasn't as plain as she thought. So now it's up to her to find out all she can about her past while looking at her own problems too.
Hanaizawa-chou Kouminkan Dayori; this one tackles an isolated town and its inhabitants' lives as the town slowly dies out. It's not linear, so you'll be jumping in the timeline; take note I guess.
Ikigami; I would recommend thinking of the main plot rather than the details. It's about many young people's last moments.
17-sai; okay so I didn't actually read this manga, but I know of the real life story it's based on and it's despair-inducing. The manga seems to be lighter, but I bet it's still horrible.
Yomawari Sensei; technically scans are ongoing, but it's episodic (and thus has no bearing on the plot) so I put it here. A series about the suffering of troubled children and teens, and the sensei that finds and tries to help them.
Hope any one of these tickle your fancy