Gin's speech about "carrying a heavy weight on a long journey" in chapter 13 can definitely induce Manly Tears.
There's a reason why this troper and her friends call Okita's sister Mitsubaaaaaawww... damn you, Gintama!
This Troper, who normally is laughing very hard whenever she watches this anime, had tears in her eyes when she saw that episode. She also thought that Kagura's fight against Abuto during the Yoshiwara Arc was very heartwrenching.
Oh God, the Mitsuba arc... This Troper was close to bawling like a baby. Seeing Okita, who is normally a deadpan snarkingsadistbreak down and cry as his sister's hand falls from his face... *bawls*
The end of the Itou Arc.
The end of episode 211 with the Four Devas of Kabukichou clashing thanks to Jirocho's attempt to take over. He and Otose knew one another in the past, and now he needs her out of the way. He appears to have killed her, in front of the grave of Otose's husband. Followed by a brilliantly done fight between him and Gin.
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 the aforementioned battle with Jirocho, where Gintoki loses.
Jirocho:Get lost, watchdog. There's nothing for you to protect here.
The reunion of Jirocho and Pirako, damn you Gintama.
There are plenty of smaller ones in the short two to three episode arcs as well. At the end of episode 108, a short story about an orphan, Kyoujirou, 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.
There's an earlier one (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.
Before that, when Sougo asks the Yorozuya to investigate the illegal underground fighting ring, the top fight 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 Tendo sect (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.
Seemingly subverted then played straight in episode 130. It starts out seeming like a slice of life story about and 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 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 is the only way they can really show affection. 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.