In Beyond the Alley of the Dolls one of the Samulacra (Dogglegangers!) finds a bunny plush, and becomes so overwhelmed by the find that he just hugs it. At the end of the episode, you can see that this clone still has the plush with him.
Since the Samulacra are clones of Sam, he probably has some predisposition to liking rabbits, since they subconsiously remind him of Max. Immediately after you watch the clone hug the toy, you cut back to Sam and Max and find Sam holding him in a Bear Hug. All together, now: D'awww!
"You'll crack me up, future Max!"
"Don't I crack you up now, Sam?"
"It was a baaad day on the moon/and then a saaad day on the moon/and an okaaay daaay on the moon/when Max came yakking back to me..."
In season three, Sam needs to be holding Max to teleport with him; so whenever you want to teleport, he grabs him in a hug, holds his ears, or pats his head, all of which are very cute to see; especially when you know that Max would probably rip off the head of anyone else who tried that.
In the finale for "Chariots of the Dogs," when they're about to go through the Bermuda Triangle, Max helps pull Sam up and smooths down his tie for him. It's easy to miss little affectionate moments like that, but they're there.
Most of the truly adorable or heartwarming moments are incredibly easy to miss, since they're just slipped in like that. For instance; in "Ice Station Santa," when Sam and Max are trying to climb up the foliage to reach the chimney of Santa's workshop, Max keeps slipping down and finally just throws himself onto Sam's back.
"You're the best friend I ever had, Sam!"
"And you're mine, little buddy."
This is actually a Fridge Logic example (I think), but if you read Sam's mind a few times in the beginning of "Beyond the Alley of the Dolls", one of the thoughts he has is something along the lines of, "Max is getting so powerful now, he probably won't need me any more..." However, Max does not seem to think so; he still follows Sam everywhere, and, like Sam says, he doesn't even need to. Now THAT'S friendship!
"There's still some of Max in there, I know it."
This one requires some explaining. Sybil Pandemik has just gone into labor after thirteen months of being pregnant. To make matters worse, she's done so inside the body of Max, now mutated into a mindless Cthulhu-esq monster. All of a sudden, the "spores", mini-aberrations that are psychically linked to Max, start chanting "Save Sybil". Even in his mutated state, and despite his usually psychotic behavior, Max still wants Sybil and her baby to be alright. To make this more amazing, mere moments ago, a part of Max's own consciousness had basically declared Max to be totally incapable of selflessness or empathy.
Sam's reunion with Max at the end of "The City That Dares Not Sleep." Well, Past Max, that is.
There's a nicely subtle moment in that scene that makes it all the stronger. Past Max is initially his hyper, boisterous self, and just skims right over the fact that he had to kill Past Sam. It seems like he was prepared to jostle into Future Sam and Max's relationship, except then he realizes that his future self is gone and Sam is really depressed. Then he just awkwardly trails off for a minute, before making the effort to perk them both up again.
The Crimefighting ending makes this even better. Upon seeing Past Max, Sam immediately sweeps up his friend into a hug. Then, after an awkward silence, the pair walks off into the sunset talking about all the crimes they're going to stop. Despite the death of their respective partners, they're more than happy to settle for a different version of them.
After Sammun-Mak has changed the universe so he is in power and Max is, and always has been, just a brain in a jar, carried on Sam's back:
Max: Hey, Sam, you won't have to carry me around anymore.
Sam: Aww, I've been carrying you for years!
In "The City That Dares Not Sleep, the Narrator taunts Sam by telling him that he'll only be able to stop him by risking the destruction of half the city. Sam responds by telling him that he doesn't care, and he'd destroy half of any city to save his best pal. He doesn't try to hide the fact that he's only doing any of this for Max's sake, and doesn't actually care about saving New York.
When talking to Little Sam and Max in "Chariots of the Dogs," they can't resist ribbing the younger versions of themselves; but Little Sam retorts that he doesn't need them, because he doesn't need anyone else but Max, who never makes fun of him.
"That's right! ...poindexter."
There's also the way Future Max looks after Future Sam, who's grown hopelessly senile from all his adventuring. Future Max doesn't show anything but compassion and understanding towards the old duffer.
A small one from 305, but when Sam tries to remind Max of the various places in the city, seeing Giant Max smile every time he remembers was pretty "aww".
In 202, when you inspect the punching bag in the office, a little back-and-forth ends with this:
Max: No way, Sam! I could never stop loving you!
It's a little less heartwarming in context; two lines ago Max had mentioned that violence was how he showed affection.
After the failed attempt to clone Max at the end of The City That Dares Not Sleep, everyone present Flint, Mr. Featherly, Stone Lincoln, and even Grandpa Stinky all seem down, showing they must have some level of affection for the deceased Max and the mourning Sam.