Batman telling Joker that there is nothing special about their relationship, that Joker means nothing to him and that there isn't even a relationship there anyway. Batman just sees him as another in a list of enemies for him to beat up (probably not even #1). Joker's face goes from shock to pure heartbreak. He doesn't even say anything, he just floats away looking utterly heartbroken and slightly shaking his head in utter disbelief. It's probably one of the only times audiences will feel sorry for the Joker across the entire Batman mythos.
The Joker's entire character arc in general could be considered a tear-jerker. For 78 years he's been waiting for Batman's validation— to be recognized as his Archenemy— only to have Batman coldly shut him down again and again. Joker's increasing desperation finally culminates in him being so depressed that he's willing to die with the city rather than go on not being acknowledged by Batman.
The Joker: No, I'm not gonna help you! At least if the city is destroyed, I'll die knowing I'm your greatest enemy. [crying] I'M GONNA HAVE TO DIE TO KNOW IT!
Batman talking to his parents through the portrait of himself and them, presumably on the night they were murdered. Despite this iteration of Batman being a goofy, exaggerated iteration, it never forgets that Batman is still an orphaned boy grieving his lost family. Even if the scene is followed by Batman dropkicking Alfred into a piano, Batman looks like he's about to start tearing up as he addresses the picture.
Before that, Batman returns home from saving the world to an almost empty house, with Alfred doing something else. It's a clear contrast to the more high-action opening of the film.
This notably shatters him when the Joker arrests himself and the other villains, with all the reporters pointing out that there's really no need for him now that the villains are locked up. His tough exterior cracks to the point where it takes him a whole week to notice Robin in his mansion.
Reporter 1: Question for Batman. Batman: Yes? Reporter 1: What are you gonna do with the rest of your life? Batman: I- Wh- What do you mean "the rest of my life"? Reporter 2: We don't need a vigilante crimefighter now that there's no more crime! You can spend some quality time with your family! Batman: My... [falls silent]
Dick Grayson gazing at some Wayne family pictures and looking like he's about to cry as he wishes he had his own family pictures, reminding everyone that, in spite of the movie's more comedic nature and his Keet personality, Dick is still an orphan here.
There's something pretty heartbreaking about the idea that the Justice League never bothered inviting Batman for celebrating their 57th anniversary. They even invited the Wonder Twins and the other original characters from the derided "Super Friends" era, but not one of their founding members, presumably because this Batman is that much of a jackass towards the other superheroes (which is more than probable, considering his constant aggressive attitude towards Superman).
Dick falls off the plane and Batman manages to catch him. Alfred falls and Batman doesn't - everything goes slow motion as everyone realizes Alfred is about to die (thankfully Barbara saves them).
The Bat-Scuttler - when Batman orders it to take his Bat-family away somewhere safe, its ears droop like a puppy and it looks sad too, before taking them away.
Batman having every Kick the Dog moment he'd done in the entire film played back to him in the Phantom Zone, causing his Jerkass Realization.
Batman is also shown Alfred, Barbara, and Dick coming back to find and rescue Batman. While otherwise heartwarming, Batman (and the viewers) get to see Dick leaving Barbara and Alfred and deciding to work alone just like Batman does. As Batman watches Dick cast aside his All-Loving Hero, Keet Robin persona in lieu of lonerNightwing, Batman is horrified as he realizes that Dick is possibly starting to slide down the same path of isolation and unfulfillment as him thanks to how he treated Dick.
Batman and Robin saying goodbye at the end when Batman tries to return to the Phantom Zone. He doesn't go but the moment is still tender. Robin tears up and Barbara's eyes start welling up.
Batman: It's gonna be okay kid. Sometimes losing people is a part of life, but that doesn't mean you stop letting them in.
It's a Fridge-Tear Jerker when you bear in mind why Alfred and Batman are Older Than They Look: Finn is playing with them. And sometimes, kids have this wide-eyed belief that life isn't short and you have up to 100 years (maybe longer, in Lego Alfred's case) to live. Pretty poignant, when you think about it.
