A precious moment between Archer and his mother, after she shoots him six times "right in the 10 ring".
Another one in Stage Two. He even shares a moment with Woodhouse. Not to mention accidentally blurting out the fact that he thinks he's in love with Lana. It may or may not be true, but it was sort of sweet when you consider Archer probably has no idea what love feels like considering his background.
As well as one with the Wee Baby Seamus. Archer makes a point of spending time with him, even though he isn't really his son, because he's "probably the closest I'll ever have." Really, "Stage Two" is a pretty heartwarming episode, at least by Archer standards.
Archer's interactions with a fellow cancer patient named Ruth. One of, if not the only time Archer has ever shown genuine niceness to a fellow human being.
Archer: "Don't you worry, Ruth; I'm gonna make them pay for it." (The "them" is the Irish mob).
Hell, his RAMPAGE against the Irish mob for selling fake cancer medication is at least 50% about them doing it to Ruth and other cancer patients. If it was just him, one could assume he would take only half the gear.
In "El Secuestro", a surprising one from Cheryl. After getting Pam kidnapped by pretending that Pam was her (they were really after Cheryl in the first place), Cheryl not only shows a lot of guilt over the matter, but refuses to sign Mallory's contract to make ISIS her personal bodyguards until Pam is rescued. Given how selfish and spiteful Cheryl usually is to her co-workers, that says a LOT, and it shows that she and Pam really are Vitriolic Best Buds.
From the two part season finale, we have Archer finally finding someone he actually loves, Malory's legitimate fears that Sterling had died and Katya giving her life to save the man that she loved.
The reveal of what Katya and Krieger were actually discussing.
Throughout the episode Woodhouse was doing all he could to stall Archer in his apartment by pretending not to know where his bug out bag is but when Archer told him that he was truly in love with Katya, Woodhouse immediately stopped stalling and gave Archer not only his bag but also his grandmothers ring. Makes seance he would be sympathetic to love no one else would understand.
Understated one in "The Honeypot" when Archer finds himself in a bar that hosts chicken-fighting matches. A big, weeping Dominican man is sitting next to him, crying over the death of his slain rooster. The next time we see Archer, he is covered in dirt and carrying a shovel, and mentions that he spent all night helping the guy bury the dead bird. For Archer of all people, this is a surprising bit of kindness.
This becomes even more Heartwarming In Hindsight after the second season reveal that alligators are one of Archer's three biggest fears (along with crocodiles, which Archer explains are also native to the Everglades).
Camp Gay: "Then where were you all night?"
Archer: "Way the Christ out in the Everglades burying some Dominican guy's rooster!"
Camp Gay: "Fun! (beat) Oh, you mean literally."
In Heart of Archness: Part 3, when Archer tells Lana that even though they fight, he considers her his friend, and his only friend at that.
Woodhouse's origin story in "The Double Deuce". All of it. The whole story was enough to impress Sterling.
He avenges his friend Reggie's death by killing the entire squad sent to kill them. He eventually ends up owning a bar in Tangier, Morocco, when a pregnant Malory Archer shows up on the run and Woodhouse delivers Sterling Archer. He even helps come up with the name Sterling.
Later, Woodhouse sells his bar to buy a boat to get to Lisbon, Portugal, and then sells the boat to get money to send Malory and baby Sterling back to the United States. He ends up raising Sterling singlehandedly until Sterling was 5.
Of course this being Archer, the sentiment is quickly undermined when it is revealed that Woodhouse kept the scalps of the Germans he killed.
In the Season 3 finale, "Space Race: Part II," Archer chooses the ISIS crew over fighting Barry.
In the same finale, Archer makes up for an incident in the prior episode. As ISIS launches the space shuttle to go back to Earth, Cyril is forced to be the pilot as he is the only qualified person remaining. In the prior episode, Cyril had flown a flight simulator until Archer disrupted his concentration. Cyril crashed in the simulator. On board the shuttle, Archer tells Cyril that Cyril is going to make it, which elicits a thank you from Cyril to Archer. AND in response, Cyril uses the shuttle's rocket propulsion to disintegrate Barry's shuttle, stranding Barry.
Yet in typical Archer fashion, the Heartwarming Moment is ruined by the end. Archer grabs the throttle from Cyril just as they're about to land, crashing the shuttle in real life. Much to the chagrin of the ISIS agents, Archer gets almost no injuries from the rough landing, while the rest of them were so severely injured that their insurance provider dropped ISIS as a client.
In "Blood Test", Mallory pouring out her heart to Archer about wanting grandchildren. Lana even tears up when she walks in, hearing the end of the conversation.
A weird meta moment in the teaser of Fugue and Riffs, which is a gentle parody of Jon Benjamin's other show, Bob's Burgers. Archer (Bob), who has no memory of his life as a spy, is closing up for the night when a four-man KGB hit squad enter the restaurant. Thinking they are robbers, offers them the money from the till, and even gives them permission to sexually assault him, so long as they don't hurt his family. Archer has been submissive to the intruders, but the moment one makes a move to murder the Bouchards, Archer—who still has no memory of his previous life, and is unarmed—brutally kills all four men.
In the episode "Once Bitten", Archer remembers back to his childhood, when he spend yet another birthday alone. Sterling's father walks into the room and gives him a stuffed alligator.
In "The Honeymooners", it's quickly ruined but Archer fearlessly diving off a skyscraper to save Lana is kind of sweet. Y'know, by Archer standards.
In "Un Chien Tangerine", Archer has a one-way confession to Kazak and almost reveals his true feelings for Lana. He gets interrupted, of course, but it's clear that he's starting to realize he might actually be in love with her, even though he doesn't know what to do about it. And judging by Lana's reaction to what she thought was a wedding ring, it might not be one-sided.
Archer asking Lana to not quit from ISIS again, because they make a good team. He even admits that he can't promise that he'll stop needling her, rather than making an empty promise.
In The Papal Chase Archer's horrifying reaction to Lana getting shot.
In "Sea Tunt Part II" Archer risking his own life, without hesitation, to guarantee Lana's survival because she's going to be a mom.
Not to mention the fact that he yelled 'I LOVE YOU, LANA!' with his last breath mere moments before drowning.
And his one condition to this event? That Lana promises to be a better mom than his own was.
And on a weird side note, Archer tickling Ray and teasing him about giving him mouth-to-mouth, especially after all the shit he's put him through was actually pretty freaking adorable all things considered. Even though it ended badly.
In "Crossing Over", Archer bonding with Jakov over shots in the safehouse. Seriously, two of the meanest, shortest-tempered characters put everything aside to make up for 35 years of lost time.
Despite everything Sterling does to him, Woodhouse was still there when he graduated from college.
Archer proposing to Lana and offering to help raise her baby in the Season 5 premiere. Granted she turns him down, but it's one of the most genuine and nice things he's done in the whole series.
Archer and Ramon remeeting well until it reveal that Ramon faked his own death bit.
In "A Debt of Honor", Archer comes extremely close to calling Ron his stepfather, though he changes at the last moment to "my mother's husband". Considering how badly Archer took his mother's marriage (and his general behavior when it comes to the men his mother gets involved with), even this is a huge step forward for him.
From the same episode, Cheryl calmly willing to put down a huge amount of money to save Pam's life is surprisingly touching. As is her happily giving over a full arsenal of family-owned weapons to the team, and then basically saving Ron's life by leading his group through an underground tunnel on the fastest possible route to the nearest hospital. For someone who is clinically insane and often sadistic, she can be surprisingly sweet!
In House Call, Lana considers running away from the mansion to avoid getting busted by Agent Holly. Archer tells her he'd support her if she took this action, simultaneously cheering her up by being his assholish self by quipping she should spend 5 minutes giving a combination "Goodbye/Thank You blowjob". She hugs him.
Lana: [tearfully sincere] I can't leave you, stupid!
And earlier on, we find out that Archer has been looking up on several pregnancy complications because he was worried about her.
Archer taking the heat for Pam for having no gun during the bus attack.
When in the same episode Pam tells Archer that the possible fatal side effects of eating cocaine are worth it because people like her now because of her weight loss he explains to her that people liked her (well, hated her less) BEFORE she lost the weight and possibly dying of an overdose isn't worth it. Keep in mind that Archer is one of the very few characters in the show that has WILLINGLY had sex with Pam on more than one occasion.
Archer going through all that trouble just to get Lana a good present for her baby shower.
Topped off by Woodhouse wheeling in Archer's old crib. A gesture that nearly makes Lana cry.
Believe it or not, Malory gets in a good one, buying Lana pretty much everything she needs for her baby. Of course, she regrets it immediately afterward when it turns out Krieger lied about his business, and she is still flat broke. But the mere fact that Malory would do that for Lana if she could afford it is nice by her standards. This is a woman who thinks a potato is a good Christmas bonus.
Lana actually being supportive of Archer is pretty good.
On a show where it's seldom very clear that any of the character even really like each other, it was kind of sweet to hear Cyril tell Ray that he finds him to be good company. Ruined by Ray pointing at Cyril and shouting QUEER.
Archer and Lana showing great concern when they think Pam was killed by Calderon. Followed quickly by Cheryl's roaring rampage of revenge.
Malory showing some rare compassion during the season 5 finale. During and after the birth, she acts downright motherly to Lana, and even acts nicer than usual to Sterling.
"Sterling Archer, I'd like you to meet your daughter."
A stunned Archer may well have missed a follow-up explanation, nearly faded away by sound effects: "Despite it all, I [Lana] really do love you."
Archer revealing that he took classes and learned how to deliver a baby to make sure that Lana was safe in a worst-case scenario (which ended up happening of course)