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Tear Jerker: Gintama
Gin's speech about "carrying a heavy weight on a long journey" in chapter 13 can definitely induce Manly Tears.
The Inugami Arc.
Early on in the series, Sougo asks the Yorozuya to investigate the illegal underground fighting ring, the top fighter of which turns out to be a kindhearted father to orphaned children who wants to try and start a new life. Just as he sets out to escape Edo and start over, the Tendoshu (who are behind the ring) have him killed. The following scenes with his last words and the children tearfully asking Gintoki to help them in exchange for their treasures (read children's trinkets) is touching in a heartwrending way.
Episode 31, when Gintoki gets amnesia from a traffic accident. Any attempts to help him fail. At the end, Yorozuya is destroyed by an incoming spaceship (Gintoki claims it's impossible for a human to stop it). Gintoki then tells Shinpachi and Kagura, more or less, "The Gintoki you knew isn't here anymore", and then leaves. Thankfully, he regains his memories by the next episode.
Another early one occurs in episode 73 about a man and a bear cub he raised, only for him to be forced to kill it later when it becomes infected by a disease and goes rampaging in a mad fit.
Mitsuba's arc in general, but especially her death, with the normally sadistic Jerkass Sougo beside her bed, desperately clutching her hand and holding it to his cheek as she dies.
Ito's death at the end of Shinsengumi Rebellion Arc.
At the end of episode 108, a short story about an orphan, Kyojiro, taken in by a group of Yakuza, he dies by the grave of the leader who he considered a father, believing until just before the end that he had failed to protect the people important to him. Of course, as soon as Gin-chan manages to convince him otherwise, he is fatally shot.
Seemingly subverted then played straight in episode 130. It starts out seeming like a slice of life story about an old dog and old man loyally dying at the same time, except the dog turns out to be an alien that eats planet and the old man was rude and told people to go die all the time. It's played straight in the end when it turns out that both don't want to die before the other because they don't want to make the other sad by leaving them alone and competing with each other while telling the other to die in the only way they can really show affection to each other. The old man dies as soon as he sees his last friend, the dog, die first after their race (while both are dying) so that he won't be left alone.
Danzo Tobita AKA Jiraia's backstory as revealed at the end of the Red Spider Arc.
The Four Devas Arc feature one in the form of Otose's supposed death at the hands of Jirocho in front of her husband's grave. It's heart-wrenching.
Otose: I have a favor to ask. They call me one of the Four Devas, but I don't have any members in my faction. My life is all you need. They serve no use at all. They just happen to be my family.
Five minutes later, we see Gintoki desperately trying to get to Otose in time—but it seems he doesn't. Cue Gintoki unleashing his rage on Jirocho, only for him to lose and crawl desperately towards the dying Otose, accompanied by flashbacks of their first meeting and his promise to protect her.
Jirocho:Get lost, watchdog. There's nothing for you to protect here.
The reunion of Jirocho and Pirako, damn you Gintama.
At the end of episode 226, when the old prisoner revealed that he knew the letters he was receiving were all written by the prison warden, his "nasty but dedicated son".
The end of the Renho Arc, which was filled to the brim with about as many shout-outs and parodies as ever, takes a sharp turn hard enough to induce severe Mood Whiplash and seemingly ends in one of these when Elizabeth left the series for good, only for it to be ultimately subverted when it's revealed that the Elizabeth leaving is a temp, and the real Elizabeth was only on vacation. NEVER FORGET THAT SORACHI IS A TROLL OF THE HIGHEST DEGREE.
In the Kintama Arc, our protagonist sudden returns to the Yorozuya after a drinking night, and he goes and salute their comrades, he finds out that he has been replaced by someone who is better than him, and the guys he thought to be his friends donít EVEN remember him. Itís not a joke. They arenít the only ones to have forgotten his own existence. And he just wants to beat the hell out of him to know a reason of why everyone forgot about him and he is just... there, taking the credit of everything he has done (over the past 40+ volumes/250+ episodes) but those who were his friends tell him that he is nuts, and that he just have to keep out of Kabukicho. Just dammit, seeing his face as Shinpachi and Kagura tell him that he CAN'T be acting that friendly on them is heartbreaking.
Oh, and it goes downhill from there. Kintoki convinces Gin that everything he lived is a lie, just his own invention... puts in doubt everything about him! It's painful!
But I must say that someone cannot cry when after losing all hope of his own existence, Sadaharu and Tama comes to his rescue, the robot maid tells him that he doesn't really have to lose his own light because it's what is differs him with Kintoki... and that he is the only one that must try to recover the memories of him with his friends.
The Courtesan of a Nation Arc has quite a few:
The 2 seconds still image of how the Bafuku sent Gintoki, Katsura, and Takasugi their beloved teacher's head after they executed him is truly riveting. After that still image, it's truly a miracle that Gintoki and Katsura end up relatively sane after the war in comparison to Takasugi, who's gone on a world-ending rampage.
The flashback of Gintoki's sensei and adoptive father being taken away from him, and how he tries to calm Gintoki by telling him that he'll be back very soon... while also asking him to promise to protect all their friends while he's gone. Made even worse when you realize that most serious arcs, all the times we have seen Gintoki risk his life for someone, especially when we didn't understand why he did it when it was just some random kid he didn't know anything about or the old courtesan in this very arc... it's because of the simple fact that he never comes back.
Maizo and Suzuran's tragic love story in general, but the end of the arc deserves special mention. They're finally reunited after so many years of being apart, only for both of them to die just as they meet again. It's not all bad, as Maizou's spirit walks with Suzuran in his arms into the afterlife, but it's still one hell of a tear jerker.
The whole damn arc with Hajime Obi "returning" and Otae's and Shinpachi's unbridled joy at having their older brother figure back. The whole thing literally ends in a giant flaming ball crafted from the tears of fans. Dammit Sorachi.