Prior to the final boss in Up Your Arsenal, the pair confront Captain Qwark for ditching the team. While Ratchet walks away disgusted, Clank attempts sympathetically to convince Qwark to rejoin them.
Clank: The people of this galaxy need you, Qwark. They believe in you. You can give them hope. You have a chance to redeem yourself and be the hero you always wanted to be...
The Future trilogy was full of this, mostly between our eponymous heroes. For example:
The end of the Wham Episode that happens on Reepor, when Ratchet and Clank reunite after having been split up. Much to the despair of many a fangirl, the bro-hug didn't happen, but you see how relieved they are to have found each other again.
For that matter, the final boss cutscene on Fastoon is a good example, too, as it borders on a platonic I Want My Beloved to Be Happy, regarding the lombaxes' new dimension, and then Ratchet's decision to stay with his friends and finally take care of Tachyon.
The ending gives us a Bookend to match the first cutscene in Polaris. The first was from Clank's point of view and the second from Ratchet's. Both include one asking if the other is alright but that isn't necessarily where this trope comes in. The latter "end" occurs after the Dimensionator was used to escape the Amazing Technicolor Battlefield that Tachyon's boss fight was held in — and the answer to "Where are we"? "We are home, Ratchet."
A Crack in Time gives us "Do you think I'd leave my best pal out here alone?" which is definitely smile-worthy the first time through, and borders on being a Tear Jerker when it's included in a Meaningful Echo later on.
What may be the best example of this in the trilogy is the rescue scene in the Valkyries' Base. Say it with me, now: awwww!
And then, of course, there is the cutscene "The Plumber's Advice", which has already been listed on the Ho Yay page, but qualifies here, as well.
On an unrelated note, the last line/monologue of A Crack in Time, helpfully provided by our resident Cool Old Guy (or a recording of him, at least), which included an oddly (and awesomely) positive Ironic Echo: "The Clock, much like time itself, is a gift — and not to be tampered with... But, like any father, my only wish is that my son does that which makes him feel whole. You are an intelligent and logical being, Clank, but intelligence and logic would have been wasted gifts without honor and loyalty; I am proud to see that you came into those on your own. So, should the Clock be too small for your plans, I pray the cosmos light the way toward a future you yourself design — and remember, the universe has a wonderful sense of humor; the trick is learning how to take a joke!"
Near the very beginning of Tools of Destruction, (directly after "Your name is Percival?") Tachyon starts badmouthing Ratchet. Though he doesn't say anything, Clank walks right up as if to defend him, which immediately draws Tachyon's attention away from Ratchet. This probably qualifies on its own, but an exchange near the end of A Crack in Time mirrors it, this time with Nefarious insulting Clank. Needless to say, Ratchet steps right in and defends him in turn. Does that make the first even moreHeartwarming in Hindsight?
In All 4 One, after Ratchet discovers Susie is an orphan, not unlike himself, he immediately agrees to help deal with the Creature Collector. And later on, when Ratchet and the others discover Susie preparing to lead an assault on Ephemeris. He talks her out of it, and promises her she can hold the perimeter instead. Her leaping to hug him and his smile are "aww" worthy.
The comic series has a handful of sweet moments — usually in passing. Lines like "I will not stand by while Ratchet is in danger." and this exchange:
Talwyn: Take care of him, Clank.
Clank: I always do.
There's also the Galactic Ranger's treatment of Ratchet throughout UYA — whereas most everybody else treated him with disdain or used him to their own advantage, the Galactic Rangers actively sought him out for help — sheepishly and even somewhat politely — and then expressed gratitude when they received it. There's even one scene where a Ranger says that everyone chipped in to get him a Map-o-matic. Ratchet returns this sentiment in the comic series, where he expresses faith in the Rangers' ability to help out with the situation on Veldin. A sentiment which they repay when, during the Final Battle, the Rangers arrive and help Ratchet by taking out Nefarious' robot mooks... And this time, they don't need your help.
A meta example: one of the hidden trophies in Ratchet & Clank Future: A Crack in Time is "Just Like Old Times". The description? Reunite with Clank.
After the end credits of Into the Nexus, the ghosts of Cronk and Zephyr, while viewing their bodies that are on display at a museum which labels them as heroes, get into yet another old-guy squabble. Eventually, they decide to go find some femdroids in the graveyard, indicating that yes, robots do have souls!