This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Mister Rogers' Neighborhood
In terms of the series itself, there are two: The goldfish episode in 1970 (#1101), and the week-long series intended to help children cope with the Cold War in 1983.
The special he made to help children understand and deal with the death of Robert Kennedy. It starts with Daniel Tiger and Lady Aberlin playing with balloons, with Daniel asking if people are like balloons that deflate once their air is all gone. After she answers he asks a new question "What does 'assasination' mean?" Oh, Daniel...
Daniel Tiger has several tearjerking moments, but Lady Aberlin always turns them into Heartwarming Moments.
An early episode (#109) from 1968 has Daniel making a gift for Mr. McFeely's birthday: a glass with his name on it. He shows it to Lady Aberlin, but accidentally breaks it. His reaction is just heartbreaking; even more so that his crying is played out very realistically. There's a full minute (minus five seconds) with no dialogue, save for the sounds of Daniel's sobs and the soft piano music in the background. Fortunately, Lady Aberlin offers to help him make a new one, and even cheers him up by saying that since he practiced with that first glass, the next one he'll make will be even better.
The episode that followed (#110) has Lady Aberlin telling Mr. McFeely the events from yesterday, and the latter looks truly surprised and sympathetic.
In episode #1292, after a few episodes of Daniel being excited about his upcoming birthday, Daniel learns that Purple Panda doesn't have a birthday (as he was made and not born.) Since Purple Panda is so sad about it, Daniel gives him his own birthday... but by the next episode, he is sick and crying with remorse because he thinks he can't ever have anymore birthdays. Fortunately Yoshi Ito explains to him that two people can have the same birthday and he starts to feel better, but it's still quite poignant that he was willing to give up something happy for himself rather than let another person be upset.
In episodes #1462, Lady Elaine extensively drills the children (Prince Tuesday, Ana Platypus, and Daniel) on getting ready for school. Thanks to her, the following episode (#1463) has a distressed and tearful Daniel under the impression that he can't go to school since he doesn't know everything. Yet again, Lady Aberlin reassures him otherwise.
Episode #1578 has Daniel wondering if he is a mistake, just because he's not like other tigers. If you really think about it, Daniel is questioning his own existence. Part of the lyrics in the song that follows are especially tearjerking. Thing is, however, this episode actually hits home with many children who had verbally abusive parents, and those kids found peace in the episode.
In episode #1483 (Big Bird's visit), a downhearted Chef Brockett visits Mister Rogers' and recalls his disappointment over a cake decorating contest he didn't win.
Henrietta Pussycat has also her share of these moments:
In episode #53, Henrietta, who wanted to play Goldilocks in the Neighborhood play of Three Bears, cries when X the Owl wants to replace the character with himself as his idol, Benjamin Franklin.
In episode #1170, she is upset because she did not have the opportunity to sing in the opera in the last episode (#1169) and was too shy to ask.
Henrietta becomes a major Woobie in the "Jealousy" week (episodes #1176-1180) from 1971, which involved Grandpere's granddaughter, Collette visiting. There were two times in the week which involved Henrietta causing an accident due to her jealousy, which led her to break down and worry about being sent away. With all of the tension she's gone through, it's DEFINITELY a good thing her friends were there to help.
In episode #1177, she knocks over Collette's picture, which is, in turn, run over by Mr. McFeely riding his bicycle. In episode #1179, she wishes the cake made for Collette would knock down. (It actually does, but due to a strong gust of wind.)
Episodes #1353 and 1354 deal with a fire breaking out inside Henrietta's house.
In episode #1027, everyone misses King Friday and Queen Saturday while they're away on their honeymoon, but especially Robert Troll with the latter. He just looks so downhearted that it's hard not to feel sorry for him.
In episode #1100 (the one before the infamous goldfish death episode), Elsie Jean cries heavily when she learns that she and her husband, Dr. Bill Platypus, are being forced by King Friday to move away, just because their baby is going to being hatched before his own.
Also, when Dr. Bill wonders why his wife is crying, he asks if the egg has cracked.
In episode #1182, X the Owl's wing gets hit by a ball and cries in pain. It's hard to see a character usually so cheery and easygoing, break down in tears.
In the following episode (#1183), Corney gets his finger pinched in a machine and (having previously questioned X's crying in the earlier episode) learns it's OK to cry.
Episode #1455 from 1975 and the sole week of shows produced in 1976 (#1456-1460) become this when viewed as the finale to the original series:
While it's not explicitly stated that these would be the last television visits, the Neighborhood of Make-Believe segment in 1455 ends with everyone heading off to remember past adventures, making reference to events that occurred in previous episodes, and Mr. Rogers subsequently discusses remembering and looking forward in a manner that has an aspect of finality to it.
#1456-1460 serves mainly as a bridge between this episode and the beginning of the repeat cycle, with Mr. Rogers and a different regular each show (Mr. McFeely, Betty Aberlin, Joe Negri, Chef Brockett and Francois Clemmons) watching old videotapes of the show in his garage (a set constructed specifically for this week) and explaining some of the Early Installment Weirdness (including the lack of distinction between the "real" neighborhood and the Neighborhood of Make-Believe, which was Retconned as the "real" neighbors pretending). There's a wistful quality to many of the scenes with Mr. Rogers and the other actors; again, while not explicitly stating the show is over, each visit is essentially a farewell appearance.
Plus, for pre-1979 fans, episode #1459 not only features the very last performance of "Tomorrow", but the only time this and "It's Such a Good Feeling" are sung in the same episode together.
In the week of episodes #1516-1520, King Friday and Queen Sara leave on a trip without Prince Tuesday. In episode #1519, Prince Tuesday wakes up crying from a nightmare he had about a lost little fish who can't find his mother.
It's hard not to feel sorry for Prince Tuesday in episode #1542, when he gets sent to his room by King Friday, for touching the latter's tricycle without permission.
Mister Rogers' visit with Koko the Gorilla (episode #1727) counts mostly as a Heartwarming moment, but when their visit comes to an end, they just spend time looking at each other, and at one point, Koko looks away, saddened and almost teary-eyed when Mister Rogers has to leave soon.
In episode #1748, when King Friday gets angry when Hula Mouse goes missing. In his anger, he breaks the toy that Lady Aberlin had bought. When he leaves, Hula Mouse, Lady Aberlin, and Lady Elaine have a few seconds of stunned silence, and Lady Aberlin almost looks as if she might cry.
In the following episode, Lady Aberlin still seems deeply hurt before King Friday tries to apologize.
LeVar Burton visiting from Reading Rainbow (episode #1725) and sharing a touching story about fathers ("The Daddy Book" by Ann Morris).
A minor one but in episode #1725 (LeVar Burton's visit), his voice breaks for a brief moment while singing the "Good Feeling" song. This actually foreshadows the second-to-last episode of the series (#1764), when this instance occurs again.
While Sesame Street taught children the educational, Mister Rogers focused on the psychological and developmental virtues of childhood. For so many of us who grew up in the late 60ís and 70ís, the success and teachings of both Sesame Street and Mister Rogersí Neighborhood went hand-in-hand and cannot be separated from one another.
Mr. McFeely: (As he comforts the tearful Read-A-Roo) Yeah, I miss him too.
Seriously. People would think this kind hearted man hurt children unintentionally in the long run. Some of the comments that come up from viewers later in the program point out exactly how ridiculous this is.
In terms of the original music written for the show, The Truth Will Make Me Free, which is about learning to not suppress negative emotions. In episode 1396, Francois Clemmons and John Costa perform a particularly beautiful version of this song that's a Tear Jerker in its own right.
It's hard not to tear up at the dedications given to John Reardon, Bob Trow, and Don Brockett after each of them passed on.
"There are all kinds of ways of remembering people. By their sounds. By their looks. And by the way you feel about them."
When the Twitch Marathon came to its finale, thousands of viewers were united in bidding farewell to Fred Rogers one last time. Tears were shed, and feels were had, as the Twitch Chat pleaded for their TV Dad not to leave, knowing that there wouldn't be a next time for Fred Rogers. Doubles as another Heartwarming Moment, for how much love and appreciation was expressed for all the good that Mister Rogers gave to the world once more. As the show does not acknowledge its own finale, it ends with Mr. Rogers still singing the normal ending song telling us that he'll be back again... which the viewer of course knows isn't true.