In "Dear Sis", when Fr. Mulcahy feels that he's useless to the unit. At the end, Hawkeye and the rest of the 4077 let him know just how important he is to them, and sing "Dona Nobis Pacem." Also in the same episode, the look on Major Winchester's face when he opens his Christmas present to see the tobogganing cap he wore as a child—a little piece of home thoughtfully sent for by Mulcahy. Dona nobis pacem.
It's even more heartwarming when you realize that it was actually Klinger who sent out for the hat and made it seem like it was Mulcahy's doing, thus helping both the chaplain and the surgeon truly connect to the holiday season.
"Potter's Retirement" has the colonel planning to hang it up after receiving the news that negative reports about his leadership are coming from inside the camp. The party in question turns out to be a mole from a general who holds a grudge against the camp, and after he's revealed we get a scene in Potter's office where Hawkeye, BJ, and Radar let him know how much his leadership is valued and beg him to stay, Hawkeye even addressing him as "Sherman".
The B plot in "Run for the Money." A wounded Private is teased by the other men in his unit and his Sergeant for his stutter, and has been called a dummy for so long that he believes it. Major Winchester rips into the men who make fun of him and tells the Private that he's seen his IQ score and that he's very intelligent, but needs to believe in himself. He then recites a list of famous people who stuttered and sends the Private off with a copy of Moby Dick, saying that it's fit for a man of his intelligence. He then retires to his tent and listens to a recorded letter from his sister back home...who stutters.
And then there's the B plot in "Death Takes a Holiday". Winchester earns the contempt of the rest of the camp by not donating to the Christmas charity drive... because, unbeknownst to them, he is instead following a family tradition of donating a large consignment of confectionery from an upmarket Boston supplier to a local orphanage. He is surprised in the act by the orphanage director, who invites him in to meet the orphans; Winchester declines, saying that the gift must be given anonymously. He later discovers at the camp Christmas party, to which the orphans are also invited, that the director immediately sold the chocolates on the black market, and he confronts the director, only to learn that the money was used to purchase enough rice and cabbage to feed the orphans for a month. He is immediately humbled by the revelation, and returns to the Swamp. Klinger overhears the conversation and follows Winchester to give him a plate of leftover food from the Christmas party... on condition that the gift must be given anonymously. They exchange a "Merry Christmas", addressing each other not as "Klinger" and "Major" but as "Max" and "Charles".
In a weird way, Charles' initial confrontation with the orphanage director was one. Charles, who's usually pretty aloof, looks about two seconds from ripping the man's arm off and beating him with it— and sounds about three seconds into the act.
Winchester seems to be part of quite a few of these. The B plot of "Sons and Bowlers" centers on Hawkeye's father going in for surgery Stateside, but the reason for his surgery (which is not identified in the episode) means his survival is not guaranteed. Winchester overhears Hawkeye trying to get through to the hospital where his father is being treated on the camp phone, and proceeds to spend the night keeping vigil with him until they can get word that the elder Dr. Pierce has made it through. As they discuss their relationships with their fathers (a discussion in which we learn that although Charles Winchester II wanted only the best for his son, he was nevertheless a typically emotionally distant patrician patriarch), Winchester calls Pierce fortunate in that while Charles had a father, Hawkeye had a dad.
Hawkeye: Charles, you've never told me anything like this before.
Charles: Actually, Hawkeye...I've never told you anything before.
The episode in which Charles meets a soldier who had been a pianist until he was wounded in the hand. Charles remembers that a composer created music which could be played with only one hand, and urges the former pianist not to give up on music, since he still had "the baton, the classroom, the pen. All my life I have wanted to play. I can play the notes, but I cannot make the music." The scene ends with the music being played, one handed.
There's plenty from the finale:
Klinger loving Soon-Li enough to stay in Korea (go back to the early episodes and you'll realize what a surprise that must have been) to help her find her family.
Hawkeye and BJ giving Potter a respectful, silent salute.
Potter giving Margaret a kiss on the forehead and hoping that her personal life will be just as good as her professional life.
Potter saying goodbye to Sophie. (Doubles as a Tear Jerker.)
Hawkeye telling BJ "I'll never be able to shake you" and the deep, desperate hug between the two of them right afterwards.
The "goodbye" stones is a two-fold Crowning Moment of Heartwarming: from the show to us and from BJ to Hawkeye, giving him the goodbye that he wanted from BJ and - it's implied - from Trapper.
Charles saying goodbye to Margaret. Previously, he had taken back a book of poetry he had given her, much to her disappointment. When it comes time to leave camp, he returns the book to her, now with a written message inside to cover, and kisses her hand, a true Officer and a Gentleman.
Speaking of Trapper, there's his exchange with Hawkeye when they think he's going home, in "Check-Up":
Hawkeye: Thanks, Trap. Trapper: What? Hawkeye: You made it bearable. I was lucky. You were honest, and open, and let me lean on you. Trapper: (smiling) No charge.
Henry's goodbyes to everyone in "Abyssinia, Henry"...before that last O.R. scene shifts everything into full-on Tear Jerker territory.
Speaking of Henry, there's an episode where he's depressed due to not being able to see his newborn son. Near the end of the episode, Radar brings in a Korean woman and her infant son, so Henry could hold the baby for a while. Edges into being a Tear Jerker, as well.
In the "Bug Out" episode, the 4077th is forced to relocate in response to rumors of the encroaching Chinese advance. As part of that, they find an abandoned schoolhouse that they believe is perfect for setting up their army hospital... except that it's already occupied by several Chinese women who are using it as a brothel. After some heated back and forth between the Madam and Colonel Potter, the ladies take an intense interest in Klinger... 's rack of fall gowns. Klinger, for the good of the unit, forfeits the entire rack of gowns, and 'buys' the schoolhouse for them. As Potter commented, "Corporal Klinger, that's the finest act of bravery I've ever witnessed."
Anytime the 4077th get a home movie from their families.
Hawkeye: Henry, if you don't give the order to cry, I will.
"Goodnight Supermen"
In "Fallen Idol" Hawkeye hands Radar his Purple Heart medal, and then gives him one of the very few genuine salutes the good Captain ever did.
When Radar is sent home, he leaves behind his beloved teddy bear, indicating both that he is now a man and that he will always be with the unit in spirit.
