Moomin's complete terror after being (temporarily) transformed by the Hobgoblin's hat. He's physically unrecognizable and his friends and family refuse to believe it's him, treating him like an imposter. He begins crying and when his mother shows up, he desperately pleads with her to recognize him. Mercifully, she does, and he changes back shortly thereafter.
The death of the squirrel in Moominland Midwinter. This scene is actually so upsetting that Tove Jansson added an author's note about how, if the reader begins crying, to take a peek towards the end of the book... where Moomin meets an identical-looking squirrel and wonders if it's the same squirrel who somehow survived anyway.
The Lighthouse-Keeper's background story, as deduced by Moomin, in Moominpappa at Sea. After finding several long-dead birds on the ground, and discovering many small graves (which Little My confirms are the graves of birds), Moomintroll understands why the Lighthouse-keeper left: The birds flew towards the light and died when hitting the glass. The Lighthouse-Keeper buried them, but more birds would eventually fly into the light... eventually, the Lighthouse-keeper couldn't take it anymore and left.
1990 animated series
Nipsu Rakastuu (The Girl With Birds). Basically, Sniff falls in love and has his heart broken. It's hard to watch the poor guy sob his heart out on the front steps of the Moomin house and not once does anyone step up and hug him.
Another episode has Moominpapa run away with Mama to seek adventures. Moomin calling out their names at the top of the hill with the sad music playing in the background makes a real tearjerking scene.
In yet another episode, Snork is awaken by a big explosion. He runs all the way to his hangar only to find it burning, which means his airship that he worked with for such a long time, is destroyed. He looks so upset you want to hug this guy.