For a gag series, Gintama has a lot of heartwarming moments. Anytime Gintoki sheds his lazy, apathetic loser persona to say some good ol' shonen lines to his friends/extras of the week.
Episode 59, where Otae gives Gintoki her favorite umbrella. The accompanying background music certainly helps.
The half-episode There's Almost a 100% Chance You’ll Forget Your Umbrella and Hate Yourself For It, in which Kagura gets an umbrella from Gintoki, ruins it, fixes it, gives it away to a pair of homeless kids stuck out in the rain. She picks up another broken umbrella, and Gintoki and Shinpachi comment on how beautiful it is.
You know, Independence Day isn't that bad, is it?
From the Yoshiwara Arc, we have Seita reuniting with his mother, then he carrying her on his back because she's been crippled by Hosen while declaring his happiness at having to carry his mother's burden.
Then all of Hyakka and Tsukuyo shows up and asks him carry their burdens at the same time, because all of them took part in protecting him, thus are acting as his mothers all alone, after said group is almost all half dead from fighting before.
In the Red Spider Arc, there's Gintoki's speech to Tsukuyo after he rescues her, about how she doesn't need to act on her own and always carry a burden alone, because she has friends she can and should rely on now.
He then follows up on it at the end of the arc, after Tsukuyo has sobered up from her drunken rampage, where he tells her to not be ashamed of the path she chose, and that her face is not ugly so long as her soul remains unscarred.
In the second half of episode 189, Kagura keeps up her radio exercises for months because of a promise she made to a friend. She tries to do so even in the rain, but she can't get her radio to work, and breaks it in frustration. Just as she's about to start crying, Gintoki steps in to help her continue.
How Madao/Hasegawa-san gave up his chance at a job in order to mend a broken family in Chapter 274.
During the Four Devas Arc, after Otose has seemingly been fatally wounded and all of Kabukicho is flaring up for war, all the past characters who Gin and the others have helped gather together to protect the supposedly disbanded and broken Yorozuya and Otose's bar.
Jirocho retiring as a Deva of Kabukicho, and returning home with Pirako at the end of the Four Devas Arc.
In 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 Kintama Arc has both Tama and Sadaharu remembering Gintoki even if everyone else has forgotten about him. The voice acting deliver in 253 really sold it.