At the very beginning, in an episode omitted from the English dub, we learn that Shingo is afraid of cats due to a childhood trauma, thus hampering Usagi's idea of adopting Luna as the family pet. After Sailor Moon dispatches of the Monster of the Week, Shingo, having spotted her and thinking she's Sailor V, asks for her autograph. She does so on the condition that he overcome his fear. Not only does he do so, but the final scene shows him happily playing with Luna.
Nephrite coming to Naru's aid when she was kidnapped by Zoicite, who thought his fake defection act was the real deal. Ironically Zoicite's actions lead to Nephrite defecting for real for Naru's sake.
Nephrite and Naru's final moments go between this and Tear Jerker.
Usagi sitting with Makoto and sharing lunch when the latter is Eating Lunch Alone because everyone else is a fraid of her.
Yuuichirou (Chad, the homeless drifter who stayed with Raye and her grandfather because he fell in love with Raye) defending Rei from a youma. "If it's for you, Miss Rei, I wouldn't mind dying!"
Sailor Moon is able to successfully de-brainwash Tuxedo Mask and their touching last moments.
The four inner senshi's spirits comforting a defeated Usagi after all dying for her (Sadly, outside of the "Sailor Moon Says" PSA segment at the end of that episode, this scene was cut in the DiC dub from the 1990s).
All of the Sailor Senshi's spirits come together and give Sailor Moon the courage to finally defeat Queen Beryl once and for all, also a moment of awesome.
Even Zoicite's final moments cross into this with Kunzite granting his last request as he dies. It's more heartwarming (and heartbreaking) in the original Japanese version than in the American dub, as the Japanese version adds that Zoicite wants to die under a shower of cherry blossom petals and the dialogue's more emotional.
Sailor Moon R
Lita donating blood to her friend, Ken, after he rescued her from the Cardian and suffered a head injury.
Naru (Molly) talking to Usagi (Serena) one last time before getting Brother Chuck'd in the later episodes, implying that she knows something about the dark crystal overrunning the city and telling her to be careful.
Ami's third focus episode, in which she is brainwashed into thinking everyone has turned against her, but reaches a moment of lucidity where she sees her friends' smiling faces and breaks the spell.
The episode where Ami almost moves away to become a doctor, and changes her mind when she gets the letter and transformation pen from her friends.
The Makaiju aliens (Ann and Alan in the English dub or Ali and En in the Japanese version) receive their second chance.
Usagi and Mamoru finally getting back together.
Saphir's death at Wiseman's hands...actually, heck, almost all of episode 86 had Heartwarming, Heroic Sacrifice and Tear Jerker moments in spades.
The final episode and this exchange between Usagi and Chibi-Usa.
Chibi-Usa: Thanks for everything, Mommy.
Also the very last scene that shows Chibi-Usa reunited with her parents and friends over a renewed Crystal Tokyo.
Neo-Queen Serenity: You saved our world, my darling. Welcome home.
The scene at the end of Chibiusa's debut episode, as Usagi tucks her into bed. Usagi recognizes that she's been through tough times just like her and says that, "If only you'd told me everything up front, I could have helped you." It speaks a lot to Usagi's compassion that no matter how much she had irritated her by invading her life Dawn Summers-style, she never would have hesitated to protect her.
Almost any scene where Usagi and Chibiusa don't fight and get along is a minor Heartwarming Moment.
To elaborate, Fiore rants that nobody else knows what it's like to feel alone, which prompts each of the Sailors to flash back to their own personal moments of loneliness: Ami was shunned because she was a bookworm and people thought she was arrogant, Makoto was feared because of nasty rumours about her being violent, Minako was thought as a stuck-up Sailor V wannabe while Rei was considered a freak for her psychic powers and devotion to the temple. Fast forward and it shows how each of them have been touched by Usagi's friendship: A scene shows Usagi taking a book away from Ami at the beach and insisting she play with them, Ami agrees. Her first meeting with Makoto is shown where Makoto asks why Usagi isn't afraid of her and Usagi ignores this and asks to share her food. Usagi also tells Minako how excited she finally is to meet Sailor V. But the crowner has to be the one between her and Rei where the two girls that are always at each other's throats just deliver the most adorable little exchange.
Usagi: Wow Rei-Chan you're a hard worker!
Rei: (smiling) Oh now you realize?
Usagi: (laughs nervously and smiles)
And the dubbed version is equally as adorable:
Serena: Rei, haven't you heard? Too much work is bad for you.
Rei: (smiling) Oh, well then you must be in perfect health.
All the time Fiore ranted that Usagi was a liar and deceiver incapable of giving anything to anyone who selfishly uses others, as he tries to rip her brooch off to stop her from using the Silver Crystal, what does Usagi do? She takes his hands and tells him You Are Not Alone. And then Fiore is shown another side to his last meeting with Mamoru when they were kids. It turned out the day Fiore left Earth was the same day Usagi's brother was born, and Usagi found Mamoru crying because his friend was leaving. Usagi gave him one of the roses she had just bought to give her mother, and Fiore learns that the rose Mamoru gave him came from Usagi. If it hadn't been for her, that moment he held so dear might never have happened. Fiore is only barely able to handle this information when Usagi, without even trying, managed to rewrite the single most important event in his life.
Fiore: Who are you?
Starts crossing into Tear Jerker right after, when Fiore takes his hand off Usagi's brooch and sees the Silver Crystal's true form - a multilayered, crystal waterlily. He starts crying, and the Kisenian Flower is destroyed by the Silver Crystal's light.
Koan's Happy Ending in episode 70, after getting her heart broken by an uncaring Rubeus. He gives her a brutal You Have Outlived Your Usefulness speech in mid-battle, even going so far as to mock her values of love, before disappearing for good. A devastated Koan goes absolutely nuts, attacking the Senshi blindly. Mars takes a hit for her to stop the carnage, and insists Koan deserves a second chance as a normal human being, which Sailor Moon provides (to her surprise) by healing her with the Silver Crystal.
And in subsequent episodes, Berthier, then Calaveras and Petz, all get their chance for the same treatment, adding up to three straight Happy Endings. Berthier's in particular is a really sweet You Are Not Alone moment.
Minako's bungling attempts to care for her sick teammates in episode 78 are strictly for comedy, but suddenly, a nice little "d'aww" moment happens out of the blue.
Usagi: Hey Minako, why do you work so hard to take care of me?
Minako: Because you're very important to me. So, I feel I should protect you.
Usagi: Why? Is it because I'm the princess?
Minako: No. I feel very comfortable when I'm with you. I like the way you're full of energy. That's why I can't leave you alone when you're sick. Get well soon, okay?
Sailor Moon S
Artemis giving Minako a bouquet of flowers after she'd been depressed for a good portion of one of her focus episodes.
Umino and Naru don't do well in the dating game contest, until he delivers a completely heartfelt speech about how much he cares for Naru and is willing to lay down his life protecting her.
In one episode Makoto loses her confidence and goes to train at a mountain temple. The others follow her and, after a pillow fight scene, Mako slips out while the others are asleep and thinks about how lucky she is to have such great friends.
Michiru, to Haruka: "You have beautiful hands!"
In the 26th episode of the S season, there is a small conversation between Chibiusa and Setsuna at the beginning. There's no set-up for it. It's just really sweet and heartwarming.
From the very beginning, almost every Chibi-Usa and Hotaru interaction is a CMOH.
Sailor Moon SuperS
Chibi-usa and Helios' developing romance in Super S is also very heartwarming. Especially the scene where Helios tries to leave out of fear of endangering her, and she holds onto him, refusing to let him disappear. Her absolutely heartbroken look and her crying pushes this into major Tear Jerker territory as well.
Chibi-usa: You said you needed me! You said you needed the power of my dreams! I won't ever let you go!
One episode has Sailor Moon nearly suffocate while fighting the Monster of the Week. Tuxedo Mask throws away all pretense of secret identity by calling out to her by her real name even in front of a civilian friend of his.
Another one has Shingo nearly drown and, as Ami begins giving him mouth-to-mouth, Tiger's Eye ask her why she's working so hard for the sake of a kid. She immediately replies that Shingo is a good friend of hers. When he comes too, she hugs him tightly while expressing how glad she is that he's okay (prompting a very heavy blush from him as he has a massive crush on her at the time).
When Mamoru needs to stay at the Hikawa shrine for a few days Diana stays with him as his "bodyguard" and calls him "Endymion-sama." He lets her know that she doesn't need to be so formal and can call him "Mamo-chan," but she declines out of respect for the person she knows he'll become in the future. He relents and invites her to take a bath with him. Her surprised expression, complete with Luminescent Blush, is adorable.
Even though Ami's father divorced her mother and left them, Ami is elated when she receives sketches from him and thinks about how he doesn't need words to send her his warmth.
The 29th episode of the "SuperS" season has Chibi-Usa draw a crudely drawn picture of Usagi, at the end of the episode when she is asked what the title of her portrait is, she responds, "The Person I Admire Most".
Noticing Makoto alone and dejected looking at a college dance, Ami offers to dance with her when Makoto says that nobody wants to dance with her.
Sailor Moon Sailor Stars
The very end of the Nehelenia arc where everything has returned to normal for her and she's reverted back to a child. She's about to go to bed when she turns around and quietly asks her servants, "Will someone please read me a bedtime story?" Then one of the unseen servants says, "Yes, it would be my pleasure."
Minako and Yaten's conversation in the backstage of the Idol Singer contest. Yaten tries to reprimand Minako for getting herself in when she should be only fighting for the universe, but Minako turns the tables on him and says that even when she wants to save the world, she should not forget her own dreams in the process. Way to go, Mina-P!
The very last episode of Sailor Stars. Specifically, the scene where Usagi, having defeated the Chaos, is floating in the skies, alone. She starts crying because she is lonely without her True Companions who were all killed by Galaxia... and suddenly, a familiar voice tells her that she is not alone... Then, after all the senshi reappeared, somebody calls out "Usako...". Usagi's reaction to Mamoru's voice and her flying into his arms, in tears...
Around the time Viz Media revealed the new English dub cast, Amanda Celine Miller, the new voice of Sailor Jupiter, was overjoyed to have gotten the part, and wrote a very touching Tumblr post about how she identified with and looked up to Sailor Jupiter as a child. It got shared all over the place, and best of all, it reachedSusan Roman, the original English voice of Sailor Jupiter. That's just too sweet for its own good.
Read any interview with the old dub cast, they clearly loved working on the show. It really is heartwarming that almost 15 years since the old dub has ended that it still means something to them