Shunrei's tearful pleas for help to a terribly battered Seiya to save Shiryu's life during the galactic tournament after they beat the living crap out of each other.
Seiya's reaction? Another tear jerker, since he immediately says that he never intended Shiryu to die, get up from the stretcher despite how he's this close to die as well, and do his best to honor Shunrei's pleas and save Shiryu. Itworks.
Same goes when Shiryu gouges his own eyes out to defeat Algol (whose weapon, the Medusa shield, petrifies you on sight and save the petrified Seiya and Shun, and they both bawl when they realize what happened. Made worse in the anime, when at the end of the episode, they're told that the doctors couldn't save Shiryu's sight, and Seiya screams in utter despair as he falls into an Heroic B.S.O.D..
Moreover, in the anime Shiryu regains his sight after literally sending Deathmask to hell. And he's in tears because Shunrei died for it. Which is followed by a tear jerker of a happier kind when we find out that thanks to Dohko, poor Shunrei's actually alive and kicking.
Lots of people decry the Ansgard Saga as pure wangstyFiller, but it has many genuinely good tear jerkers. Thor's backstory and his first encounter with Princess Hilda, as well as his Famous Last Words when he aknowledges his loss and begs Seiya to help the Brainwashed and Crazy Hilda return to her senses. Fenrir's backstory as a whole; Freya attempting to make Hagen pull a Heel–Face Turn and failing as well as her reaction to his death (even Hyoga cries); Volker's Suicide by Cop at the hands of Mime and Mime's reaction when he finds out the truth about him; Sid and Bud's Cain and Abel backstory as well as Sid telling Bud that he always knew they were Separated at Birth and always loved him, ultimately to die together frozen in an embrace when they abandon the palace; Siegfried's Heroic Sacrifice for his beloved Hilda from the Bronze Saints and Poseidon's Siren Sorrento, Seiya's Heroic B.S.O.D. when he thinks he killed her, Hilda recalling her Mind Rape at the hands of the Nibelung Ring as she invokes Odin's power...
Capricorn Shura's death was deeply sad. Watch the episode and weep, if only for his commendably tragic Heroic Sacrifice after realising everything he'd ever believed in was a lie.
Also, Shaka's Heroic Sacrifice and using the last bit of energy to send a message to Saori. She breaks down in tears.
On that note, Saga's, Camus' and Shura's first Athena's Exclamation—they knew they'd be shunned by the rest of the Saints, but they did it anyway to complete their mission. They resigned their honor as Saints, to do what they had to do. I teared the hell up, man.
Sanctuary arc. Seiya and Shiryu. Seiya and Shiryu feel Shun's cosmo explode, and then they realize that Shun has pretty much sacrificed his life to save Hyoga's. They're appalled and almost in tears, and so are the viewers. And even more tearfully, in the anime Hyoga arrives in the middle of the Curb-Stomp Battle that Milo is giving them, with Shun's comatose body in his arms, and explains what happened. Then he urges them to get up, go to the next Temple and carry Shun with them, which they do. All my tears, and all the audience's.
In regards to the last point... the next Temple? Sagittarius's. The Sagittarius cloth subjects Seiya, Shiryu, Hyoga and the now awake Shun to a Secret Test of Character, and once they pass it, they find Aiolos's last will... one in which he asks all of the Saints of the future to take care of Athena. If you can ignore the way their tears are animated, it's one of the most beautiful and heartbreaking moments ever of the series.
In the Hades arc, Shun throttled Hades from inside his body so Ikki can kill the god along with him. Ikki's flashback of his cry-baby brother growing up into a saint fitting for Andromeda's name as he tearfully attacked him is heartbreaking.
The death of Pandora on the Hades arc in front of Ikki's gaze; it's worse by the fact that they are beginning to get chummy; keep in consideration that this is something that 1. Never happens with Ikki; and 2. Happens to a girl that is beautiful. Damn! none of those guys is going to be able to have a girlfriend EVER!
Don't forget that Ikki had loved and lost before with Esmeralda, making this doubly frustrating.
When Ikki is summoned to Hades' chamber and has the opportunity to kill Hades himself by killing his host body, which is Shun, Ikki is actually shown to hesitate. Ikki, who doesn't have many restrictions on killing unlike the other Saints; Ikki, who had no problem killing a perfect illusion of his brother in the Poseidon Arc, and was also fine with digging into Shun and taking out Hades in an illusion, hesitated when he was given the chance to kill his own brother. Hades even mocks him for this before defeating him.
The entire series is a cavalcade of constant sadness; the mere fact that Athena puts herself in harm's way is precisely because she doesn't want her Saints to be hurt in the first place. Everything that happens after that is because she always refrains from fighting back at all... which is why she always needs the Saints' aid when she gets kidnapped. Judging by the Tenkai Hen Overture, Saori is just content to be able to take care of Seiya on her own.
Which is kinda messed up, considering that she treated them as dogs when they were just kids.