This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Dragons: Riders of Berk
In light of the sequel... all of the father/son moments between Hiccup and Stoick. All of them.
Becomes worse in the third season which airs after the second movie but takes place before it.
The sad looks on the dragons faces in "How to Start a Dragon Academy" when the kids were about to lock them up.
In "Viking for Hire" when it looks like Hookfang will have to be put down, Snoutlout can only manage to repeat what Hiccup says, with more emphasis and tears.
Hiccup: Hookfang's a good dragon.
Snoutlout: (On the brink of tears) He's a good dragon!
The scene from "In Dragons We Trust" when everyone has to leave their dragons on the island. Everyone's scene is sad, but of course it's Hiccup and Toothless' goodbye that's the worst. Hiccup takes off the saddle, and is about to leave, but Toothless keeps following him. He holds his hand out as a gesture to stop... and then Toothless moves forward to touch it, reflecting the moment they bonded in the movie. Hiccup, obviously thinking the same thing, moves forward to hug his best friend one more time before leaving. His expression through the whole thing... And not to mention that "Where's Hiccup" is playing in the background.
"I'll be back for you. I promise."
Special mentions should also go to the Twins who switch from one head to the other to say goodbye showing that unlike everything else they share the dragon and most likely switch heads to ride.
In "Portrait of Hiccup as a Buff Young Man", Stoick digging through the rubble to get to Hiccup. Think about it; this is the second time he's thought Hiccup was dead because of something he did. This time it was an innocent moment of insensitivity, but it still caused Hiccup to go out and try to prove himself. Now imagine the guilt.
The moment itself is pretty bad too. Hiccup doesn't usually show it directly, but he's a huge "Well Done, Son!" Guy - and as this episode proves, his very un-Viking physique does still bother him - and his reaction over the painting is heartbreaking.
Hiccup: "It's not me."
Stoick: "Sure it is! It's you but... you know... bigger! Stronger!"
Mulch: "Now that's the son of a Viking Chief!" (he and Stoick agree and laugh, oblivious to the implication or Hiccup's reaction)
<Hiccup says nothing, but wears an absolutely crushed look on his face and just backs away, staring sadly at the painting>
Watching the dragons get sick in "Dragon Flower" was hard, especially when we found out that the allergic reaction to the flowers was of the deadly sort.
Hiccup saying goodbye to Toothless (yet again; the fanbase cannot handle them being separated) before going off to get the venom that will save his life.
And then after all of that where Hiccup is back with Toothless, he tells his worried father that everything is going to be fine. Yet immediately after his father leaves he worriedly hold back onto Toothless. This showing that despite Hiccup's statement, he was not fully sure himself.
Seeing the look on Snotlout's face, first when his father lectures him about not losing, and later when you see the look on his face when he knows he has all but lost (though quickly reversed thanks to Hiccup deciding to throw the race), you can't help but feel some pity for him, as the source of his Jerk Jock tendencies become apparent.
Hiccup's increasingly desperate attempts to protect/save Toothless in "When Lightning Strikes".
More than that, the way he ran after the wagon when the villagers overtook them and chained Toothless. He's desperate to keep up, but he can't with his prosthetic. It's just painful.
Toothless throughout the episode: His reaction on seeing the villagers hunting him - after everything he's done for them - is heartbreaking, not to mention his terror when they chain him up, and of course his utter panic when Hiccup is struck by lightning and almost drowns. Fortunately they seem to realize they've gone too far and by the end spoil Toothless with endless fish and apologies.
Seeing Hiccup be struck by lightning was pretty hard to watch.
The strain on Hiccup and Toothless's friendship in "What Flies Beneath" was also very hard to watch. Toothless was caught up in his grudge against the Whispering Death to the point where he refused to let Hiccup help, even though doing so left both of them flightless and handicapped. It culminated in Toothless firing a plasma blast just inches from Hiccup's feet to keep him from following, and Hiccup had to resort to jumping off of a cliff just to snap him out of it.
Seeing the twins, usually are Comedy Relief, fighting with each other in a way that isn't comical at all. And an easy thing to understand for anyone who's ever had brothers or sisters; Tuffnut's rant about not having anything that's just his, as opposed to theirs, hits particularly hard.
It's a small moment but you see Snotlout's insecurities come through in "The Defiant One" when after they've managed to survive the giant waterspout and wondering where Hookfang is, you hear him say "I always knew Hookfang didn't like me." Couple that with the small speech he gives later to Hiccup which indicates how overshadowed he feels in comparison you almost feel for the guy. Almost.
Which then comes back in "Breakneck Bog" where he says it again.
Watching poor Toothless trussed up and refusing to eat or drink anything.
In "We Are The Family", Mildew (Yes Mildew) shows off some Tear Jerker scenes. First of all he has Fungus, the only thing he actually cares about, taken from him and suggested that he was going to be eaten. He then helps Hiccup get back to Toothless. He even performs a Heroic Sacrifice at the end asking Gobber to make sure that Hiccup knows he is sorry for all he is done before letting go so they could escape. Then another type of Tear Jerker scene comes when he never changed and was working with Alvin all along, showing a character that finally seemed to be turning a new leaf is even worse.
Hiccup talking about how he was afraid of forgetting his mom in "Breakneck Bog".
"Race to Fireworm Island". Dear God, even with Like You Would Really Do It in full swing, the possibility that the Fireworm Queen might have killed Hookfang will get to you.
Even for those who regard him as The Scrappy, it's hard to not feel for Snotlout. The whole tone plays out that he is breaking down because he is likely the very cause for what could have been his best friend's death and goes from haughty jerk to near-sobbing wreck. Anyone who lost a pet would empathize with him.
"Fright of Passage"; the intro is a flashback to a young Astrid seeing her uncle go after the Flightmare. He dies off-screen, frozen in terror or more accurately, paralyzed by the mist.
The attempted sensitivity of the townsfolk make it worse as the others tell a freaked out Viking to not say it in front of her.
A minor one for Snotlout in the same episode. The twins send him on a series of wild goose chases before they'll let him into their party, but by the time he's done it's over and everyone's gone home. You can hear him tearing up over the whole thing.
The ending of "Big! Bang! Boom!" has Stoick reluctantly freeing Thornado so he can protect a group of adolescent Thunderdrums from the wild dragons on Dragon Island.
Toothless' reaction to finding the dragon bones in "Dragon Eye of the Beholder, Part One" below decks of The Reaper. Followed by Hiccup comforting him and apologizing that Toothless had to see them.
A bit of a Cry for the Devil moment in "Dragon Eye of the Beholder, Part One" but it's revealed that Dagur has been imprisoned on Outcast Island for three years, with Alvin as his jailer. Even though Dagur committed regicide and is The Caligula, that's a wee bit harsh.
Gothi in "Eye of the Beholder, Part Two" has obvious trauma related to the bite mark on her arm that matches the marks on the Dragon Eye, and for good reason: when searching for glacial water with healing properties as a teenager, she lost her search party who are killed onscreen by a Snow Wraith. Gothi survived but she makes it clear she never wants to go back to that island. It's a great Kick the Son of a Bitch scene when she whacks the Snow Wraith several times and obtains its tooth with her stick.
Ah man, Heather has not had a good life in the three years since we last saw her, and it's just getting worse.