The entire two-part "Make a Wish" and "Attack of the Octobot", after a fight with Doc Ock goes haywire. Pete, feeling more down than usual starts to consider giving up being Spider-Man, until Madame Web introduces him to his biggest fan: a sweet little girl named Taina. While the pair go out for a little harmless web-slinging fun, Spidey fights the Doc again and gets amnesia, and the ''good'' doctor manages to convince poor Spidey that they are a criminal team. Taina, believing herself at least partially to blame for this, spends the rest of the episode trying to help Pete get his memories back. But the best part comes at the end, when she asks her hero to reveal his Secret Identityand he does. More heart-wrenching is that she is revealed to be terminally ill.
When Norman Osborn saves his son Harry against a falling beam, when Hobgoblin fires a pumpkin bomb at him. He survives, but it makes Harry realise how much Norman cares for him.
In the episodes "Framed" and "The Man Without Fear", Peter Parker is framed by the Kingpin for stealing defense secrets and is defended in court by Matt Murdock (Daredevil). Murdock is a very famous and expensive lawyer in this continuity and he refuses to say who payed him to defend Peter. At the end we find out that J. Jonah Jameson footed all of Peter's court costs anonymously. He'd always liked and respected Peter and had believed in his innocence the whole time, but was too embarrassed to show it publicly. He helped him despite him quitting the Daily Bugle in the first place.
The very end of the show, where Madame Web, who has made Spidey's life miserable with her enigmatic tests and dragging him to the Secret Wars, reveals that she's going to help him find Mary Jane and says that after all he's had to put up with he "deserves some happiness". It was very sweet coming from a character who was often equal parts enigmatic mystic and deadpan snarker to show a warmer side.
The end of "Mutants' Revenge," before the X-Men depart.
Cyclops: I'm sorry we got off to such a rocky start. You're a heck of a team player. Gambit: You got that right! A regular ace! Wolverine: Nice workin' with ya, kid. Jubilee: Real nice. Spider-Man: Thanks. I don't know what to say. Wolverine: Just remember, no matter what problems ya got or mistakes you've made, ya don't have to carry the load by yourself. Ya got friends if you need 'em.
FlashThompson comforts Debra Whitman when the Vulture rapidly ages her. What makes it especially frightening for them/heartwarming for us, is that Flash and Debra have no idea what just happened, and for all they knew, her aging was permanent. And he still comforts her. He's also the one who calls her out when she starts really throwing her life away after Morbius is lost for good. He really does care about her.
During the "Partners in Danger" season we re-meet Scorpion, and it turns out he's laying low and trying to save up so he can have his mutation reversed and marry his sweetheart. In his last appearance he's back to crimes, yes, but implies he snatched the rings to marry her.
Also, when he is put in danger in said episode, his girlfriend actually comes to his rescue through the aid of the Vulture. It's touching that she cares so much to risk her own life for his sake, and that the Vulture would also help Scorpion despite their relationship bordering on the vitriolic.
During the final episode of the "Six American Heroes" arc, "Enter Electro", Captain America does a Heroic Sacrifice to take him and the Red Skull back into the vortex, before which he gives amazing parting words to Spider-Man to never give up:
Spider-Man: Cap, I can't do this without you! Captain America: You can do it, Spider-Man. You're a true hero — First, last, and always! Fight on, Spider-Man! (Cap wrestles the Red Skull and himself into the vortex)
During "Guilty" we see Jameson being his snarky self as usual about Peter refusing to take Spider-man's picture, but when it settles down, Robbie discusses a personal matter of him leaving to see his son play in the All Stars. We get this exchange which is one of the rarer times Jameson shows genuine heart. As well as the moments why Jameson and Robbie are friends. As well as why he personally tries to clear Robbie's name in a recent frame up.
Jameson: You know, I miss those days with my own son. You should cherish these times with your family. You're a lucky man.
Robbie: That's one thing we can agree on, I am a lucky man.
Spider-Man defeats Spider-Carnage, an Alternate Universe insane version of himself by bringing the still-alive Uncle Ben from another universe to talk Spider-Carnage out of it. The speech moves him so much he snaps out of it long enough to jump into a void dimension. Then Spider-Man gets to talk to Uncle Ben for one loving moment.
And then this is followed up by Spider-Man meeting Stan Lee. "I've always wanted to be appreciated as a real hero. It seems you've made me into one."
On a meta-level, although Stan Lee and Madame Web don't quite interact on-screen, they voice compliments about the other ("Who is that exotic lady?" and "I think he truly is special"). As Joan Lee, Stan Lee's wife, voices Madame Web, this was intentional.