Anytime the number of dudes and dudettes who come back is the same as the number who left base a few hours ago.
Even better: complete a mission, especially the more difficult ones, without any injury whatsoever. There's nothing like imagining the squad being cheered as they come back home, tired but satisfied and able to dismiss any paramedics sent to aid them. Even the base personnel comments on, essentially, how well you're taking care of your soldiers.
Breathing that sigh of relief when a soldier in critical condition is stabilized. They're badly hurt and will probably be out of combat for weeks... but they're alive, and that's what matters.
Alternatively, beating the mission while you don't have anyone that can stabilize. If you beat the mission during the time the soldier is bleeding out, he'll be considered as stabilized and taken back to base alive. You can just imagine your soldiers fighting not just to defeat the enemy, but to save their fallen comrade.
Watching wounded troopers joking around in the infirmary. Knowing that they made it out alive and will be back on their feet is wonderful.
In the opening cutscene to the first Terror mission, one of the fleeing civilians stops to pull a wounded man out of the way while under plasma fire. It's a shame that he gets eviscerated by a Chryssalid shortly after that...
An odd example, but while the Uber Ethereal is quick to criticize most of the alien slave races during the final mission, it's surprisingly complimentary toward the Thin Men.
Having been assigned two soldiers with the same last name, it's always heartwarming to think that there's a couple or siblings out there trying to save the world together. Bonus points if one of them becomes The Volunteer.
When a soldier is either killed or knocked out, and one of your soldiers panics and kills the alien who took out their buddy. It's heartwarming to think that maybe that shot was payback.
On the same note, the Support class can obtain a skill which allows them to trigger Reaction Shots when the enemy fires, not just when they move. It's not unheard of for a teammate to get suppressed by an alien, call for backup and immediately see something along the lines of "Col. Hill fires a reaction shot", and have said suppressing alien be gunned down.
The Memorial Wall, combining this with a Tear Jerker. It could have been just a plaque on the wall, or a notice board. In this game, its surrounded by shot glasses, with a few pictures posted up on the wall, starting out as a few wallet-size clippings if you've only lost one or two and growing to a full size remembrance with almost none of the wall uncovered if you're sending troops into the meat grinder. All the while, somber Taps-esque military music plays in the background. It is really touching that in-universe, your soldiers aren't simply anonymous weapons sent into a meat grinder, but comrades who remember their fallen dead. Gone, but not forgotten.
Even more so, with the above in mind: visiting a memorial wall with only a few pictures on it. The fallen are remembered, but it feels good to make sure that wall is kept sparse, and that the men and women under your command are kept alive and well. It's good to remember those who have died in the line of duty, but better that there are many more who can do the remembering.
Skipping the tutorial and very careful play can keep that wall empty. If you haven't lost anyone, the option to look at the memorial wall is greyed out and unclickable. You're pretty much expected to lose at least some people in the game... but you haven't.
Rarely, sometimes the name generator will actually come up with first or last names of your own friends in real life.
Whenever you succeed in thwarting a Terror mission, the panic level across the entire continent drops sharply. The aliens specifically set out to induce hysteria, but XCOM's stopping them actually made the populace feel more at ease.
Combined with awesome as, depending on how many civilians you save, the better the panic reduction. Letting no more than four die will not only result in an Excellent rank, but you'll also be looking at a panic drop of at least 3 in the country where the attack took place, and a further panic drop of at least 1 across the rest of the continent. For the aliens' point of view, they sent their deadliest and most lethal units to horrify humanity... And their troops fail so horribly that, if anything, the attack had the exact opposite outcome of what they wanted.
A meta moment of heartwarming is when you consider who the Council members are. These are countries that to all appearances hate each other - Argentina and the United Kingdom, China and Japan, the United States and Russia - but they created XCOM together because they consider the continued survival and freedom of the human race more important than their personal squabbles.
It gets even more poignant with the troops they send, when you realize that XCOM HQ must be a lot like the Olympic village - a lot of different countries sending their best and brightest to kick ass. And instead of competing with each other, these men and women are willing to fight and die for someone from a country that their homeland is often at odds with. Or maybe they are competing with each other, to keep the team focused and battle-ready. You never know...
A subtle one in Enemy Within is with the design of the MEC Augmentation system. The whole reason for the useless (in combat) Base Augments is explained in their description. Their first function is to give the MEC Trooper mobility while out of the MECs. The second, more important reason? To ensure that, after their sacrifice, they will have a chance of a normal life after the war is over. More willing to experiment with alien technology and eugenics or not, the R&D team still haven't lost their moral compass.
There are three - well, technically two - distinct named soldiers in the regular campaign with Enemy Within: Delta-Two/Argentine Heavy, Zhang, and Durand. All three have high Will stats, and all have a good chance of being psychic and becoming the Volunteer. If this happens, all three have a very poignant end to their tales:
Delta Two survived first contact, and he survived every other battle up until that moment to reach the Temple Ship. And after watching countless other XCOM troops die under his command, he finally ends the war he saw begin, and stays behind to rejoin the rest of his fallen comrades while saving the world.
Zhang, a weary man who has served the underworld, committed countless crimes, and killed men before. He abandons his old life and old friends for an uncertain future and a common enemy. And at the very end, he redeems himself for his past crimes by giving his life to save all mankind.
Durand, an ordinary woman with extraordinary abilities, who was tortured and experimented on to use those powers. With XCOM, she finds a chance for revenge, and sees the end of the monsters who ruined her, and she eventually acts to save the world from being destroyed by those same monsters.
For extra points, her intel can help save the lives of three fellow abductees. These three 'Furies' will turn out to be grateful for your efforts and pledge their help to save humanity from the alien invasions.
For maximum heartwarming points, bring a team consisting of Delta Two, Zhang, Durand, and the Furies on the final mission. For all of these troops, it's very much a personal fight, and they will finish the mission.
A minor touch to the "Ant Farm" view of XCOM HQ is that you can actually look in multiple rooms of the Barracks, and see your soldiers doing all sorts of things with each other in their downtime: they exchange jokey banter, exercise to keep in shape, and visit injured comrades on their sickbeds. It's clear that the XCOM operatives don't see each other as interchangeable RedShirts - they're friends forged by fire, comrades in arms, and a true Band of Brothers, ready to defend their squad mates, and their planet, at all costs.
There's the obvious Doylist reason of not crippling a player, but if a country withdraws from the XCOM program, the soldiers you have from that country don't leave with them. Whether it's out of loyalty to their friends in the program or because they know they have to see the job through, it's hard not to feel like they're effectively saying Screw the Rules, I'm Doing What's Right! and essentially going rogue from their own national forces to keep fighting the good fight, no matter what it might cost them in the long run.