Just about everything in the second episode of the first OVA.
Screw that, the whole first OVA series is a big Tear Jerker. And the original manga/TV series had its moments as well.
The first TV episode. Kenshin has had enough pretending not to be the Battousai, and shows for the first time why he has such a strange philosophy to fighting:
Kenshin: "A sword is a weapon. The art of swordsmanship is learning how to kill. That is the truth. What Miss Kaoru says is sweet and innocent talk that only those whose hands have never been stained with the blood of men can believe. But to tell you the truth, I much prefer Miss Kaoru's sweet and innocent talk over the truth, that I do."
As for The Movie itself? There are several: Gentatsu's last moments and Shigure's reaction, Shigure's goodbye to Toki, Yahiko's huge Heroic BSOD when he finds his Big Brother Mentor Musashino's lifeless body among those of Shigure's followers, and especially Shigure's death in Kenshin's arms.
Say what you want about Seisouhen, but admit it. You cried at least once during either episode.
Kenshin and Kaoru die. Yes, it's ultimately considered Canon Discontinuity by Watsuki and many fans for good reason, but it doesn't take away from how heartbreaking it is to see Kenshin and Kaoru using what little strength they have in their bodies to see each other one last time, ending with Kenshin dying peacefully on Kaoru's lap.
The OVA did have genuinely sad moments, like the last time Kaoru and Kenshin make love before he leaves and share a very sad pillow talk, Megumi trying to urge the deathly ill Kaoru to live a bit more so she can see Kenshin a last time, or Sanosuke doing the same to an amnesiac Kenshin in China.
Soujirou's steady breakdown during his fight with Kenshin is so heartbreaking.
One can't help but tear up when he pleads with Kenshin: "If you claim to protect the weak, then where were you when I needed you?!"
To elaborate, Soujirou underwent major abuse at the hands of his family due to being the bastard child of his father and his mother (who was his mistress). He also endured their scathing remarks about him being a bastard child and we learn that he was only allowed in the family for appearances sake. It isn't until Shishio turns up badly wounded that Soujirou eventually becomes his disciple and kills his entire family, resulting in a Heroic BSOD that he didn't realize had, until his fight with Kenshin. It really puts some perspective on that kinda stuff.
Kenshin's farewell to Kaoru before he goes after Shishio. No wonder poor Kaoru got depressed (if perhaps excessively so) after that.
Surpassed by Kenshin's backstory from the Rememberance arc (animated as the Tsuiokuhen/Trust and Betrayal OVA) in which Kenshin accidentally kills his wife Tomoe after being temporarily blinded, by a group of ninjas. She was trying to help him and to not lose her second chance at happiness, and then SLASH. The shock of this causes Kenshin to swear to never kill again, and Tomoe's brother Enishi to swear revenge and to turn Kenshin's life into a living hell.
In the series, there is also Yumi's death in Shishio's arms after he fatally stabs her with his own katana and then comforts her as she passes away, and Kenshin's first visit to Tomoe's grave in years. Both of them with the Adagio Cantabile of Beethoven'sPathetique Sonata as background music.
The manga scene where Kenshin finds Kaoru's (supposed) corpse and completely breaks down as he believes Enishi has finally gotten his revenge on him.
The end of the first Tsukiokuhen OVA, depicting how Kenshin and Tomoe met.
Sayo's death. God, poor Sayo. The Shimabara arc was the last good part of the anime, and despite having a Bittersweet Ending, Sayo's fate remains a Tear Jerker.
The flashback where Shogo and Sayo watch their father die fighting the Shogun's forces, then told by their mortally-sick mother to leave her and "become strong as iron", shortly afterwards seeing her shot dead on the beach, and then sailing away with their uncle... past the crucified bodies of everyone in their village. And then young Shogo made an oath to become "as strong as God!" and the source of his messianic complex was made painfully obvious.
When the Oniwabanshuu died protecting their leader. And God, the look on Aoshi's face...!
Anji Yukyuzan's horribly depressing backstory. My God, Anji's screams of fury and pain when he sees the shrine burned down and his stepchildren dead, especially the teenage girl with a Precocious Crush on him... it never fails to bring the water works for me.
The Pocket Protector that saves him from Sanosuke's first successful blow... is a memorial tablet dedicated to them. He's been carrying it all these years. And he will not stop slaughtering the Meiji Government until he feels that the world is a safe place for them to be reincarnated into.
Anji: "Now do you understand?! With just prayers and wishes, not one thing can be saved—NOT ONE THING! Until those children follow the circle of reincarnation and they are reborn into this world... Myouou no Anji shall NEVER LOSE!!"
Taku Iwasaki's heartbreakingly gentle and heartfelt musical score for the ending of Rurouni Kenshin: Memories.
Hell his score for all of the OVA's counts as this.
One of the last images from the Trust and Betrayal OVA: Kenshin sits under a tree during a storm clutching the shawl of Tomoe, who recently died. As he does that, Tomoe's ghost wraps her arms around him to try to comfort him.
The scene between Enishi and Geezer at the end of the Revenge arc. Enishi, after being defeated by Kenshin and left emotionally broken, ends up in the same village Kenshin went to after he (Kenshin) was left emotionally broken by the thought that Kaoru had died by Enishi's hands, and with the same broken look Kenshin had while sitting there. Then, he meets with Geezer, who is heavily implied to be his and Tomoe's father, but they only remember each other from when Enishi was a little kid. The two end up talking about how they "remember each other from a distant past", but without revealing their names or anything that could tell each other that they were indeed son and father. However, it's Geezer words that make it all the more special, with the old man saying that Enishi's place was not there, that he, like Kenshin, would one day recover and continue to live on, but that he should just rest in the meanwhile.
Kiyosato Akira. All he wanted to do was build a name for himself and become a man worthy of his fiancee. Unfortunately the samurai he is guarding is targeted by Kenshin. The entire time during his very short duel with Kenshin he is frantically thinking "I don't want to die! I can't die here!" but he can't back down from the fight and ends up dying senselessly. The senseless part is more apparent in the first OVA, when the very samurai he is guarding actually shields him from a fatal blow while screaming "You can't die here!".
All of the flashbacks showing Kenshin during his time as Battousai. It's a testament to Takeru Sato's acting that, despite him technically playing someone (Kenshin-as-Battosai) 7-9 years younger than himself (Sato himself is 23, Kenshin began killing people at age 14), he was able to portray a young man who can effortlessly kill yet is beginning to be crushed by the weightof his crimes, however well-meaning they might be.
Kiyosato Akira, getting back on his feet even as Kenshin carves him up, screaming and gurgling about someone important to him. Even when he fell for the last time, he simply refused to die, whispering it over and over and crawling across the ground, until he is given the coup de grāce. The next day, we're shown his fiancee, most definitely Tomoe, breaking down in front of his blood and rain-soaked corpse.
This flashback itself was triggered by Kenshin seeing the corpses of dead policemen Jin-e rampaged against in his outings as the fake Battousai, ending with Kenshin's self-reproach (which shows how unresolved his issues with his past are as of yet):
Kenshin:"You are still a killer."
Saito's subdued yet-no-less icy reproach to Kenshin after looking on at the aforementioned dead cops, strewn together with the thugs Kenshin knocked out in Kaoru's dojo earlier in the film, just shows how much he actually cares for his colleagues and subordinates.
Saito:"If you killed just one of them, my policemen would still be alive."
Speaking of Saito: He never shows it, but the words he consistently repeats to Kenshin (from the prologue at the Battle of Toba-Fushimi and at his last scene) betrays a man embittered by the years, yet still animated by his own code of justice to push on even in a world he doesn't appreciate:
Saito:"Live by the sword, die by the sword. There's no other path for us."
Kenshin ruminating over the agony of the innocent civilians who were poisoned by Kanryu, and realizes it is his inaction that caused their suffering; this finally drives him to bring the drug-dealer to justice.
However odd this might sound, considering the backdrop above, we can find Jin-e's life to be one the more you think about his background in this film. While he has always been thought of as a bloodthirsty bastard in the manga/anime, the film supposedly shows him to us in the prologue as someone who has been warped by the violence of the past era, who perpetuates his violent way of life because it's the only thing he knows. His use of the name "Battousai", in a way, becomes comprehensible, if all the more tragic. He could have been what Kenshin have become had he not learned the value of people around him.