This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Hey Arnold!
The Bittersweet Ending of "Pigeon Man". Pigeon Man has to leave, because his home was destroyed by Harold and his friends, but he thanks Arnold for teaching him that some people can be trusted, and he hopes that there'll be another Arnold wherever he ends up.
The life of Pigeon Man in general is sad. Heck, the rings on his outfit that his pigeons grab onto can be seen from the start of the episode. From the beginning, he knew he would have to do this someday...
The flashback to The Fall of Saigon at the start of Arnold's Christmas; where Mr. Hyunh gives up his daughter to a soldier in a helicopter so she can have a better life.
When Arnold sits on a bench with Gerald, lamenting his failure at giving Mr. Hyunh his Christmas miracle.
The end where Mr. Hyunh finally reunites with his long-lost daughter (and the fact that Helga gave up the Nancy Spumoni boots her parents got her for Christmas just to pay Mr. Bailey to find Mr. Hyunh's daughter and help Arnold believe in miracles again) is more tearjerking almost any other moment on this list.
Helga revealing how much her family life sucks on "Helga on the Couch." Earlier in the series, it was somewhat funny how Helga's family life was a mess (her father was a blowhard who cared more about his beeper store than his family, her mom gave up being a parent and has become a covert alcoholic, and her college-aged sister moved out so she can have a better life—despite that she's painfully naïve to the darker side of humanity)... but when Helga reveals that she's been ignored since she was preschool-aged in favor of Olga, any humor that could be mined from it evaporated.
One of the MANY MANY MANY heart-breaking details about Helga is that she IS smart. Heck, she’s BRILLIANT! She’s well versed in the arts, she can and does read WELL beyond the fourth grade level and writes volumes and volumes of poetry. She could have won the spelling bee if she didn’t intentionally take a dive to stick it to her dad and do Arnold a solid. Her anonymous poem won the school wide poetry contest. She’s remarkably intelligent and talented!…But how many times do we hear her compare herself to her sister or to Lila as “Okay so maybe I’m not smart, pretty, funny, popular or nice like Lila/Olga?” Despite the fact that she IS just as smart if not smarter than the both of them? How many times do we hear her compare herself to her best friend Phoebe - who if we are honest, will admit is more STUDIOUS than intelligent. And how many times do we hear her say “Maybe if I was prettier/nicer/smarter Arnold would like-like me back?” Helga was raised being constantly compared to her older sister. She’s been told so many times, directly and indirectly that she’ll never be as good or worthy of attention and affection as Olga. And there’s a part of her that believes this. She’s a 9 year old girl - and she believes she’s never going to be smart enough to be noticed. That’s why she resorts to bullying tactics, because negative attention is better than no attention at all.
There's also a brief moment at the very end where Helga hugs Dr. Bliss.
"Parents Day," where Jerkass Big Bob thoughtlessly calls Arnold "orphan-boy" at the family event - sending him into a deep depression over his parents (the writers try to preserve some humor with Arnold going through a comical Heroic B.S.O.D. after Bob's comment; most viewers probably find that it didn't take). Later, Arnold asks Phil for the true story about his parents. Lots of fun and heartwarming moments, until the final flashback where baby Arnold says goodbye to his departing parents for what turns out to be the last time. And that dream he has after hearing the story.
"I can't find my mommy and daddy."
The situation is also pretty bad for Phil because he has to try to explain what happened to his son and daughter-in-law.
Arnold: What happened to them, Grandpa? Phil:(solemnly) They never came back; never heard from them again.
Arnold: What about their plane?
Phil: It was never found.
The scene at dinner, where Arnold, after overhearing Bob Pataki refer to him as "orphan boy", explains that he doesn't want to go back to the tournament, as it's only for children and their parents, and while he DOES love his grandparents, they aren't actually his parents. Even Arnold's grandma stops acting insane when she realizes how serious the situation has become.
Made even more heart-wrenching by the fact that Helga, who rarely shows that she genuinely cares about anyone's feelings and is usually apathetic to Big Bob's jerkass tendencies, immediately admonishes him for the "Orphan boy" comment. Even the girl who tortures Arnold on a regular basis (albeit to hide the fact that she has feelings for him) was legitimately saddened and upset by what Bob said, and rushes to apologize to Arnold for her father's thoughtlessness.
The beginning of "The Journal"; when Arnold is feeling down on the anniversary of his parents' disappearance, so much that he tries to give up any hope that they will ever return.
The end also qualifies. Arnold's father wrote on the last page of his journal that their trip to San Lorenzo would be "quick" and they "would be back soon". The way the family reacts as Phil reads the last page of the journal. Dang.
Several moments involving Helga's family:
In "Quantity Time," when Bob looks legitimately hurt after Helga tells him to leave her alone after he took her out for a pony ride because he thought she liked it (and thought she was "Seven... eight?" years old).
Miriam has a similar reaction in "Road Trip", when Helga tells her she's a lousy mom, and she should take her of her 9-year-old daughter.
The most shining example is "Arnold's Thanksgiving," when Bob and Miriam are panicking, since they thought Helga had gone missing. After Helga walked in, they also expressed their thankfulness for Helga as a daughter, which made Helga smile (which she rarely does in her own house).
Lila may be widely disliked in a lot of fan circles, but her situation in her first appearance (where she's living in a rundown apartment, her mom isn't around, and her father is unemployed and desperately looking for a job to support his daughter) was heartwrenching.
Add to that Helga, Phoebe, Rhonda, Nadine and Sheena bullying her and making her cry. Really, Lila was a flat-out Woobie in her debut.
The episode "Curly's Girl," in which Rhonda repays a debt to Curly by pretending to be his girlfriend for five days. She dumps him as soon as they agreed it would be over, and everyone hates Rhonda (even her best friend Nadine) for breaking his heart. To win back her popularity, Rhonda tells Curly to pretend to be his girlfriend again, but then HE dumps HER after hearing her in the bathroom talk about how disgusting she thinks he is (after he forcibly kissed her). As a result, people feel sorry for Rhonda and she's popular again. Then Rhonda goes to Curly's house to thank him (she thinks it was an act), and he's sitting on his front steps with his glasses off (for perhaps the only time ever in the series), eyes wide and full of tears. He didn't look like the little psychotic freak he always is— instead, he's just a depressed little boy with a shattered heart due to his unrequited love.
Despite how much of a jerk he can be, it's pretty sad seeing Sid's meltdown in "Monkeyman!" after two muggers stole his boots.
The part in "Chocolate Boy" where viewers find out he's obsessed with chocolate because his nanny, who always gave him chocolate, left when he was little. He cries piteously about this while hugging Arnold.
If you're a cat-lover, "Harold's Kitty" has some Tear Jerker moments, made more so because Harold was a loner and a bully at the time. The real Tear Jerker, however, occurs when he has to give his kitten back to its real owner; he comes out of the house holding it and crying, in front of the whole neighborhood. Awww.
Ends on a heartwarming note, though, when the cat's owner (an irritable elderly woman who previously hated Harold) decides to let him visit the cat out of gratitude for taking such good care of him.
In the Episode "Mugged", Arnold learns Karate to defend himself from muggers and to help others from being hurt. However, his incredible Karate skills go to his head, and halfway through the episode he attacks a random guy who just walked up to Arnold wanting directions to a bus stop. Not only is the man now pantless, he's crying uncontrollably and slowly walks off, constantly repeating "I was just trying to ask you where's the bus stop". A small crowd looks at Arnold with complete disgust as he realizes what he's done.
A blink-and-you-miss-it moment from the same episode: While Arnold is showing off his crueler side at school, Helga catches a glimpse of it. She's absolutely horrified.
In "Weighing Harold," it seems like everybody is suddenly making nasty comments about Harold's weight. His weight loss cruise fails because he was afraid the other participants would think he was fat, and ate to counteract the stress. If that's not enough of a Tear Jerker, Arnold volunteers to help Harold lose weight, but Harold initially can't do it. He breaks down crying while explaining to Arnold that everyone around him has been saying he was fat, decimating his confidence.
Worse this problem happens in real life, as some people who try to lose weight get mocked when they run in the park.
For anyone who has ever been bullied or used, the final scene in "Phoebe Skips," which shows Phoebe crying in the girls' room, is a Tear Jerker.
"Grandpa's Sister" has the title character visit the boarding house, and the two of them are constantly at each others' throats. Arnold helps rekindle their relationship by getting to the heart of the problem: As children, they had a pet Scottish Terrier named Pooter, who they loved dearly. One day he got out of the yard and was run over by a milk truck. And for 70 years (71) Phil and Mitzi blamed the other for Pooter's death.
What Helga's nanny says to her at the end of "Helga and the Nanny", after Helga frames her and gets her fired. She really read Helga like a map.
Nanny: You're such an angry girl, Helga. And you won't let anyone help you. So you have to live with your unhappiness.
"The Flood"; Helga falling into the water when the class is trying to save Mr. Simmons is one thing. Hearing her scream Arnold's name and Arnold's reaction to seeing her fall is another thing entirely.
The ending of "Helga's Parrot": While it was necessary to preserve Helga's secret crush on Arnold, seeing the parrot get eaten by a monitor lizard was a combination of one of scariest and saddest moments in the series. The latter is easily shown with Arnold, who, having grown attached to the bird, had a heartbroken stare on his face while Lilah comforted him.
The Christmas Episode. So many things to nominate, so little time... The ending may well require a box of tissues, depending on your constitution.
Arnold's neighbor Mr. Hyunh, trying to escape the Vietnam War with his toddler daughter Mai, ran to the American embassy only to find one more person was allowed on the last helicopter going to America. After giving up Mai to the Americans, he came to Arnold's city to find her but couldn't do it. At the end, as everyone is exchanging Christmas gifts and Arnold feels terrible for not being able to find her, the doorbell rings...to reveal Mai, who was found by one of Arnold's friends.
When Helga finally gets the perfect present combined with the love and attention from her parents (instead of the usual jerkass filled home life), she realises that she has to give it up to re-unite Mr Hyuh with his daughter and then she realises, after spending all that time trying to find the present that would make Arnold fall in love with her, that this would be the greatest thing she could do for them all and make them happiest. So she gives it all up and takes no credit for it, giving them a proper Christmas miracle and yet again ends up with her love for Arnold unrecognised, standing alone in the snow.
The episode where Olga is engaged and Helga finds out that the fiance is a Jerk Ass. Helga finding empathy for Olga, whom she's always envied and pushed away because of that, always gets me.
She even called him out, then told him to just walk away. AND THEN she forged a note from him to prevent her sister's heart from breaking more than she knew it would...*sniff* BE STRONG, BE STRONG!
It's a bit dark, but Arnold attacking the guy looking for a bus stop in "Mugged". The poor guy was beaten and humiliated just for trying to ask for help and looking a little intimidating. Arnold's shame and embarrassment and the man's sadness and confusion are just so relatable.
"The Longest Monday", an episode about the 4th graders trying to escape the 5th graders' trashing of them as part of an annual bullying tradition, ends with Arnold trying to talk sense into Wolfgang and his squad of bullies about ending this cruel tradition...and he gets tossed in the trash anyway like his classmates.
In Stinky Goes Hollywood, Stinky, among hundreds of kids (and Oskar) that applied, is picked as the Yahoo Soda kid and instantly gets famous. He's genuinely happy and believes he's found something that he's good at. Then he and Arnold overhear the managers saying they only picked Stinky because of how dopey he talks. Stinky's crushed, and ultimately decides to keep his integrity by refusing a million dollar contract, despite his family and peers dismayed and distraught of the decision. The only person who knows the truth and sympathizes with his choice is Arnold.
In a tears of rage-inducing way, the climax of Arnold Betrays Iggy. One can only imagine all the kids watching the show who felt for sure that things would work out for the best and Arnold would avoid needless public humiliation at the last minute only to be horrified and possibly scarred to see things not play out that way at all, instead being treated to a gut-wrenching, drawn out scene of Arnold walking down a red carpet out of his home and back, all while unsettling somber music plays, crowds of people laugh cruelly at Arnold's humiliation (that's being broadcast on live TV as well), and Iggy has a My God, What Have I Done? meltdown, which proves not enough for him to be redeemed or forgiven for what he brought about, by both Arnold and the audience.