The idea of Homare exploring Lava Cave because of his innate heat resistance from his Playing with Fire powers. The tearjerker came mainly from how other characters reacted to this.
Azuma stayed behind and what did he have to do? Wait for Homare's return. This already happened in the past with his family, and his family never returned. Now it was clear that Homare would be carrying out a dangerous mission, meaning there was a chance that he might not return at all.
Hisoka's was no better. Hisoka was barely containing his emotions, and when Homare almost fell off a cliff near Drill Tunnel, he just lost it and punched Homare, telling him to not do anything stupid and that he'd better return. Homare telling that he would so that he could continue taking care of Hisoka on August's behalf just cinched it.
Homare's decision to isolate himself in Lava Cave after evolving to Vritramon and being unable to stop himself from slaughtering DeathMeramon. He was afraid that he'd end up hurting everybody if he didn't do so.
The whole Guy vs Homare sequence. Homare, or rather, his uncontrollable Beast Form, tried to kill Guy. Guy kept trying to calm Homare down and help him gain control, but whatever he did, he failed.
Made even tearjerking when you realize that hurting his loved ones was the last thing Homare would ever do, as in the game, he had his issues with being a broken cyborg and inadvertently hurting everyone else with his words. Now he was physically hurting Guy and trying to kill him. All while he helplessly watched everything.
It got From Bad to Worse when Tsumugi later started a discussion about this. Homare would never want to hurt his friends, but this in turn, caused him to panic (which was a no-no in controlling Beast Form) when he couldn't control his Beast Form and ended up trying to kill Guy.
The battle ended, but with Homare sustaining serious injuries and a cardiac arrest. Guy imagined how Homare's death would snowball and hurt many people in the troupe, which caused him to cry and take a chance with CPR, hoping that it'd save Homare's life. It thankfully worked, but Homare got his ribs broken in process and was still dying.
Hisoka's reaction at Homare's broken body. Remember Homare saying that he'd continue taking care of Hisoka on August's behalf? Hisoka was tearfully begging Homare to not die, because if Homare died, it'd feel like losing August again.
Tasuku and Hisoka then discussed how they'd never been nice towards Homare. It was especially heartbreaking when Hisoka admitted that he'd been an Ungrateful Bastard all this long and didn't deserve Homare. Made even more heartbreaking that Hisoka was actually not giving himself more credit.
The misunderstanding between Guy and Hisoka. Hisoka was clearly upset at Guy nearly killing Homare, all while Guy never intended to hurt Homare in the first place. But they couldn't even speak up and settle their problems. Not that Tasuku could help them, as much as he wanted to.
Azuma fumbling in his work. Piyomon scolded him for it, not knowing that the recent happening was pushing Azuma's Trauma Button hard.
When Guy was about to spill the tea about the whole Lava Cave incident, Azuma tried to listen to everything but his resolve slowly crumbled as the story progressed. By the time Guy was about to talk about the cardiac arrest, Azuma couldn't take it anymore and left.
The whole incident was so bad that it had Azuma assume his Safety in Indifference nature again.
Unimon's story when Tasuku and Guy visited Plotmon's grave. Living in the literal Death Mountain Mount Panorama with Plotmon forced it to adopt Safety in Indifference philosophy, so that when either it or Plotmon would be killed by the disasters, neither would get hurt too much. Then Plotmon died and left Unimon alone. It turned out to hurt just that much, as Unimon could never forget about Plotmon. Unimon decided to cherish the very few memories it had about Plotmon, but the lack of interaction caused a lack of memories, so all it could do was building a grave for Plotmon. While Unimon felt some closure, it also felt a lot of regret.
According to Piyomon, Unimon actually came to File City and worked in the shop as an attempt to forget about Plotmon. Like Azuma, it blundered a lot because it got too much things in its mind, and Piyomon kept scolding it.
More on the heartwarming side, but still pretty tearjerking: Hisoka explaining how he lived on with painful memories. Memories, good or bad, were equally precious to him, because his family lived there as long as he remembered.
When Homare finally wakes up from his coma, Hisoka sheds many Tears of Joy. All his hard work paid off, but Hisoka continued to work hard, be it relieving Homare's pain, giving him emotional support, etc. It is as heartwarming as it is tearjerking.
Homare's gradual descend to the realization that his word retrieval ability was impaired. Throughout the narrative he had trouble naming objects, but it became especially apparent when Azuma suggested that he composed a poem or two. He knew the word he wanted to say, but couldn't quite bring it out to the table and said it to be "on the tip of his tongue." It put him in a state of denial and caused him to panic. And it only got worse: his injuries hurt more as he panicked. With some convincing from Tsumugi and Hisoka, he eventually decided to rest and accept that the impairment was incurable and had taken away his ability to compose poems.
When Hisoka spoonfed Homare a bowl of porridge, Homare's lack of appetite caused him to only eat a quarter of it before asking to save it for later. It reminded Hisoka of himself, but worse: never finishing his meal nor saving it for later, which often led to Homare scolding and urging him to eat more. Hisoka felt extremely guilty.
Homare saying that he would always prefer Hisoka as his caretaker broke Hisoka so much. Homare clearly appreciated Hisoka's efforts and was not afraid to shower praises and thanks. Similar to the spoonfeeding scene, it reminded Hisoka of himself, but a complete opposite of it: rarely appreciating Homare's efforts.
Tasuku seeing Homare do crossword puzzles is a heartbreaking Dramatic Irony when he said that it should be a piece of cake for Homare, a poet. He was not aware that Homare was doing crossword puzzles to help him with his word retrieval ability.
The whole revelation moment. Homare was clearly having trouble recalling words and Tasuku witnessed it, but Tasuku kept denying it and even said that he refused to believe it. With some convincing Tasuku finally accepted it and just went to hug Homare tightly, and felt that Homare had lost some weight and was distraught. Tasuku, who usually argued with Homare, felt incredibly sorry for Homare and really wanted to hold him close.
Guy breaking down when Tsumugi told him the cause of Homare's Curse of Babel: brain damage. Guy was the one who inflicted cardiac arrest and did CPR on Homare; in other words, Guy was the one who caused the Curse of Babel. And Guy never intended to injure Homare so badly that his heart stopped; rather, Guy was forced to do so because he had no choice back in Lava Cave. Tsumugi had a good reason of hiding this from Guy when he and Tasuku were away at Beetle Land.
