Several instances, but "Marooned," "Gone, but not Forgotten" and "Change of Address" (the final episode) especially.
"Just a Dream Away" from the second episode...
In "Gone..." the lyrics are particularly profound: Once I wasn't here, and then I suddenly appeared... just like water flowing, I know where I'm going... you got to leave, to stay.
"Change of Address" is one giant case from beginning to end. You'll likely be reduced to mush long before, but Sprocket's whimpering at Doc's words (and Doc's words themselves) that they'll never see Gobo again is the definite breaking point. Of course, mere moments later, they do see Gobo, but it doesn't make the emotional impact any less. And the whole "Magic Be With You"' segment with the tunnel will break you up to and including the very end of the credits.
Gobo's song "Petal of a Rose."
The song "Goodbye" from the episode "Manny's Land of Carpets": "Goodbye, goodbye, every eye is dry. Leaving? There's no grieving just a rainbow in the sky. Goodbye, goodbye, goodbye."
"Mokey's Funeral" manages to evoke this even when we know from the beginning that Mokey's actually okay.
Red: It couldn't be Mokey. She's my best friend.
Gobo: No, it was her, I'm sure of it.
Red: But you don't understand. She's my BEST FRIEND!!!
Junior's song, "And the Joke Isn't Funny Anymore" certainly speaks to this event.
There's an early episode where Gobo is off on some adventure or other overnight, leaving his roommate Wembly alone for (presumably) the first time. The song Wembly sings will haunt you forever.
If "The River of Life" doesn't make you sniffle at least once, you have no heart. Boober stands at the hole to Outer Space, begging the Silly Creatures to stop poisoning the caves, asking if the Fraggles or Gorgs have done anything to upset them. What Boober doesn't know is that Doc and Sprocket are out at the momen, and he's speaking to an empty, uncaring room - except for us.
Outside of the series, one line uttered in an interview that Gerry Parkes gave after the series ended: "I miss my Sprocket."
The episode where Red realizes that all of her talents just don't seem to be as wonderful or important as Mokey's. And instead of becoming angry or spiteful, she craves to be Mokey. The song "Dreaming of Someone" is truly heartbreaking.
The Trash Heap's attendants' interpretation of the situation makes it doubly heartbreaking when they discuss how she thinks everyone likes Mokey more than her, "...especially Gobo!"
"Marooned", especially the way Mokey holds on to Boober's scarf (it fell off when he and Red were trapped in the cave) throughout the episode, as well as Red and Boober's little moments in the cave.
Pretty much everything about this show, now that Gerry Parkes (Doc) has passed away. The last two episodes, "The Honk of Honks" and "Change of Address", might be the worst, since both episodes featured Gobo finally meeting Doc after five seasonsnote technically four, but the last season was split into two smaller seasons due to Executive Meddling. Keep in mind that Jerry Nelson (Gobo; along with Pa Gorg, The Trash Heap and many others) had passed away two years earlier, in 2012; to see their characters onscreen together is almost too much to bear.