Any time Apollo thought about or expressed his own suppressed guilt over Daphne and Hyacinthus. In addition to that, when he expresses his desire for eternal companionship/marriage, but saying that it's not in his destiny.
The Hidden Oracle
Apollo tells Percy that one of the previous times he was turned mortal was for killing the Cyclopes who created the lightning bolt that killed one of his demigod children. He admits later in the book, after seeing the way Nero has manipulated and abused Meg into loving him while also fearing his wrath, that it wasn't an accidental or irrational decision. He compares the way Nero talks about his "Beast" side to the way Zeus would talk about his lightning bolt before punishing Apollo with it, and says that he hated the lightning bolt because it was easier than hating his father.
Meg's betrayal. It utterly breaks Apollo at that point in the book and while she ultimately does the right thing, Nero's hold on her is too strong to break at this point.
The Dark Prophecy
Flashbacks to Apollos relationship with Commodus. You know it's going to end in tears when they're So Happy Together, but it's still tragic no matter how evil the latter turned out to be.
Apollo killing Commodus is this, full stop.
The death of Heloise the gryphon, made even more heartbreaking considering that she has just laid an egg, so her mate has to raise their child alone.
The Burning Maze
The pure desperation the dryads are in as they realize that they are living on a time bomb, their Palm Springs home soon to be dissolved into fire and dust. Under Grover, they manage to gather as many survivors as they could, but their numbers are dwindling just from sending search parties alone. Of course, Apollo can't help but snark a little regarding a bunch of anthropomorphic cacti worrying about the end of the world (mostly at Aloe Vera's expense), but he understands that their situation is unfair.
One of the clues to Herophile brings up Hyacinthus. Apollo doesn't take it well.
The death of Jason Grace. He doesn't technically accomplish much, not going out in a blaze of glory, and he is dead before he can say a proper goodbye to any of his friends. Rick twists the screw even tighter at the end when Leo arrives. He is devastated that he Never Got to Say Goodbye, we already saw Annabeth's reaction at the end of Ship of the Dead (which is horrifying in context), and one can only imagine how Thalia will react to the news once she finds out...
What we do see however is Leo's reaction, which is just as heartbreaking as one might imagine.
Then Leo tells Apollo and the others this before going off with Piper.
"Go on, you guys. Take... take Jason home. Do right by him."
What's even worse to imagine is the other Seven's reaction to Jason's death. Especially if you consider Reyna and Nico, Reyna who saw a great leader and comrade in him. While Jason was the first person Nico came out to, and maybe even his first friend (or at least the first without serious emotional baggage involved).
Keep in mind, one of Nico's powers would mean he felt that Jason was dead, just as he did at the end of the last series with Leo. Same applies to Hazel.
Also, this is probably the biggest deaths, emotional-impact-wise, in one of Riordan's works. For the first time, this is not a side character or villain dying to teach the protagonist something. This isn't so much a lesson as just a brutal death, and it is one of the protagonists, someone we've followed for several books, a frequent POV character, being completely removed from the picture.
So much for Jason telling his and Piper's grandchildren the story of how they beat the Giants...
Apollo doesn't even have more than single day to befriend Crest and he has to personally watch him get stabbed multiple times by Medea. His narration makes it clear that he takes it as hard as when the above Jason died.
"Crest was no longer listening. His eyes had turned glassy. His tortured muscles relaxed. His body crumbled, collapsing inward until the ukulele sat on a pile of dusta small, sad monument to my many failures."
The River Styx taunting Apollo in his sleep, and when she refers to Crest as just another pandos, Apollo yells, "His name is Crest!"
Its worse if you remember the descriptions of Tartarus in the previous series. All monsters reform there after they die, but Tartarus is so nightmarish and bleak its would be a fate worse than death if you didn't get a chance to leave it through the doors of death. Which Crest might not be able to, as its implied its hard for nice monsters to escape.
The Tyrants Tomb
Tarquin really puts Apollo through the wringer by confronting him with some of his as he puts it Greatest hits of dreadfulness with Koroniss ravens, Harpocrates and the Sibyl of Cumae.
Koronis was the mother of Asclepius. Ravens were once white and peaceful like doves but incredible gossips. They blabbed to him one day that Koronis was cheating on him, in a fit of rage he killed her but saved their unborn child, and for their role blasted the ravens turning them black and made them into the ill tempered mischievous birds they are today.
Harpocrates was a Ptolemaic god who was rather funny looking who Apollo basically acted like the stereotypical playground bully towards.
The Sibyl of Cumae was a pretty young woman who had pledged eternal service to Apollo. Unfortunately he fell in love with her. The Sibyl was not interested and scared because she knew about what tended to happen to his lovers, she accidentally made a wish to have as many years as the number of grains of sand she had in her hands when she had picked them up and he granted it and when she refused him for the last time he angrily cursed her that while shed live that long she would age. Eventually her body withered away leaving just her voice in a glass jar.
He feels incredible guilt for all of this.
Its revealed that the bodies in the Tiber without count couldnt be counted because they disappeared, Tarquin turned them into Vyralokai, the Ancient Greek version of zombies.
Hazel's reaction to Frank's apparent death. After a hard-fought battle with too many deaths already, she has to suppress her grief for yet another death that hits too close to home. The best she can do is take out her anger on her enemies and send Arion to retrieve Frank's body.
Despite the entire legion honouring him for saving the day, Apollo still feels unworthy of their praise, as so many had sacrificed themselves just to ensure his survival and the success of his mission.