"Richie Almost Dies," from midway through the fifth season, was intended as a tearjerker. Richie is involved in an accident while test-driving a motorcycle, and suffers a severe concussion, one that puts him in a coma. Howard and Marion are told that their son might not live, which leads into the two tearjerking scenes:
Leather Tuscadero (Suzi Quartro), the sister of Fonzie's girlfriend Pinkie, plays the piano — the tender ballad "Find Strength From Your Friends" as a montage of clips featuring Richie plays. The Cunninghams and other regulars sit sadly in the living room ...
... except for Fonzie, who has gone to the hospital to visit Richie. Fonzie is horrified at the sight of his best friend lying unconscious, and scared at the thought that he might not make it. Fonzie goes to a corner of the room and tearfully pleads with God to let Richie live ... before having a complete emotional breakdown.note While this was supposed to be a tearjerker episode, latter-day critics frequently lampooned the episode as an example of 1970s Narm, dismissing the "it's OK to cry" message by citing (primarily) the over-the-top script and acting, and inclusion of a very late-70s-sounding pop tune (in an episode set in 1959).
Richie after his return from service in the Korean War. Gone is the carefree Audience Surrogate of the early days of the show, he's now emotionally unbalanced and its implied he has PTSD from his experience. Sure, he hides it well, but its clear he's not the boy he was.
It can also be funny, but the moment in "Fearless Fonzarelli", when Fonzie says "Remember the good old days when I wore a leather jacket and everyone thought I was cool?" could be tearjerking for viewers who have nostalgia for those days.
Learning that Joanie's white mouse Herbert got eaten could be hard for sensitive animal lovers.
The episode "The Physical" is mostly a lighthearted episode about the boys getting their draft notices, but the tag is a surprisingly serious and poignant moment where Marion worries about what might happen if the boys actually have to go to war, and Howard unconvincingly tries to reassure her that the country won't send more young men to die overseas. Then there is a moment where they sit silently and listen to the voices of the characters playing outside... until the tension is broken with a quick joke and the obligatory reprise of the theme song.
The death of legendary showrunner Garry Marshall. He died of pneumonia brought upon by multiple health issues he suffered following a stroke several years previously.
The sudden death of Erin Moran, the actress who played Joanie. Her unexpected passing from stage four cancer at the not-exactly-old age of 56 was an especially devastating blow to her co-stars, friends, and fans of the series.