"The major tenants of the show are sarcasm, cruelty and violence. Affection and warm, fuzzy things, anything that would inspire "Awwww!" ...it's not in the show!"
—Jhonen Vasquez on why this page really shouldn't exist
Walk of Doom. Zim actually shows genuine kindness to Gir. It's a really minor case and wouldn't even be noticeable other than that...well it's Zim.
Pretty much any time Zim is ever nice to anybody.
A lot of Zim's CMOH seem to relate to Gir, for instance: "I already promised the moon to GIR."
Not to mention Zim telling Gir "please" that one time.
And then there's Gir's obvious affection for Zim, like in "Germs": "Aww, somebody needs a hug!"
Not to mention his line in the Halloween episode: "I only hope poor GIR is alright by himself...against those zombies!" The fact that he's a supposedly self-centered and unfeeling alien who actually really cares about his defective little robot, well...awww.
In an unfinished episode, Zim reacts to GIR appearing to have been turned... Well, you know, just listen to it. This short scene is also a bit of a Tear Jerker.
One of the earliest examples is in Battle Of The Planets, when Zim sends Gir to stop Dib. "Maybe that little robot isn't such a bad evil minion after all..."
The unfinished episode, Mopiness of Doom. In a nutshell, Dib and Zim realize how much they need their rivalry. It ends with them shouting insults and death threats at one another, both as happy as can be.
When Dib actually finds competent help from Nasaplace's janitor in "Battle of the Planets"...
Being a fan of Big Brother Instinct, this troper found Dib's mercilessly one-sided compassion for his sister in "Gaz, Taster of Pork" almost unnaturally endearing, considering how the show usually is. Not only did he go to incredible lengths to help her remove the curse, but he was also literally begging Shadow Hog to take the "punishment" for "failing" the trial on himself instead of her and fighting tooth-and-nail to get inside the fortress to help her. I mean, sure, part of it was to save his own skin after she threatened him, but it's clear the safety and well-being of his sister was his first priority.
I always found Zim as a smeet saying "I love you cold unfeeling robot arm" strangely heartwarming.
Now, we know Zim as, basically, a total jerkass. But then somebody created this post on their blog and it makes you think that maybe he isn't as bad as we think.
"Bad, Bad Rubber Piggy" was a particularly cold-blooded episode in which Zim's time travel machine keeps making Dib have debilitating accidents years before they'll even meet—by the time he's eight, he's got an oxygen tank, liverspots, a mechanical claw for a hand, lethargic, brain damage... and then Zim goes all the way and KILLS HIM... so Professor Membrane brings Dib back to life with a Humongous Mecha powersuit so that he can't be injured ever again! Given his normal treatment of his "insane" son, that's probably the nicest thing we've ever seen him do for the boy. Who cares if it's an alternate timeline? It's still sweet to see.
That same timeline also had a Gaz who, while not even close to the "sweet little sister" stereotype, was just a little less evil to Dib. Explained the accidents right away, sounded a bit concerned when he didn't automatically remember... for her, that's downright affectionate. Pummeled!Dib's life may have been filled with horrible accidents and rubber piggies, but at least his family cared for him a bit.
Gaz cares so much about spending time with her dad that she goes out of her way to save Dib from being experimented on by Zim. Not because she cares about Dib, mind you, but because she's afraid that if she brings home Dib as a horrible mutant their dad will cancel dinner.
It is kind of admirable that Zim still tries to prove himself as an Invader despite the Tallests' blatant hatred of him.
Despite all the crap Zim puts him through, Keef never gives up on his mission to be his friend or loses his cheerful demeanor.
Dib stargazing at the end of "Tak, The Hideous New Girl," saying how wonderful it'd be to just "fly up there and see it all." It's a small validation that Dib really isn't just an attention-hungry nut job, he really is fascinated with alien life.
Probably the only sincerely nice moment in the entire show is Dib's speech at the end of "The Most Horrible X-Mas Ever," where he encouraged everyone to stop buying into the spectacle of Santa and just go home to be with their families. It even works... at least until ZIM, dressed as the Easter Platypus, buys the crowd off with shrimp in exchange for Dib's beating.