In Season 3, Zach and Flo escape elimination for the first time due to a non-elimination leg, and Flo lets loose some of the most heartfelt words about her teammate, about how he stayed positive no matter what despite her giving up. Even if she was seen as one of the most useless and annoying people to have ever run the race, these words were especially heart wrenching.
Similarly to Flo, Ian was seen as somewhat of a blowhard during the Season 3. He constantly bickered with his wife Teri, but when they both won the penultimate leg, Ian recites some genuinely sweet words to Phil about her, saying that Teri was not only his lover, but his friend.
In Season 4, Reichen and Chip's Fast Forward meeting the world's most adorable orang-utans.
In Season 5, after running an impressive race at that point, but ultimately coming in last, Charla and Mirna end their race by thanking Phil for the experience, and Charla talked about how she showed the world what she could do.
In Season 6, after being eliminated from the race, Aaron says he had been waiting until the finish line to propose to Hayden, so he does it there. Even though they didn't win the race, they won each other.
Season 12's finale. Between Rachel declaring T.K. her favorite person, Ronald finding first place in his daughter Christina's heart, and the cheers of "Grandpa! Grandpa!" for Nicholas and Donald, the entire Final 3 had heartwarming moments to go around and made it seem that they were all winners.
Season 14 had the racers face a difficult final roadblock which involved searching 300 surfboards with pictures and place 11 of them featuring an event from each leg in order (with the pictures being not immediately obvious). Luke and Jaime were both missing one each and were getting increasingly frustrated figuring out which one it was. Despite fighting for second place, Luke shows Jaime the one she's missing at his mother Margie's advice, and Jaime in turn reveals the completed order to him. The two of them had a close bond throughout the race, and despite both of them having been shown as intense competitors, in the end, their friendship turned out to be far more important than their final standings.
Upon getting eliminated for losing a passport in Season 15, Zev (who has Asperger's) went out, not by saying, "I showed the world what disadvantaged people could do", but, "I showed myself what I could do". And again in Unfinished Business, he and Justin decided to soak in the scenery in Brazil when it became clear they were done for, with him claiming the experience was worth more than a million dollars despite how often the race took him out of his comfort zone.
In Season 17, Nick and Vicki go through some heavy arguing in Ghana, with Vicki being on the receiving end of a "Shut Up!" and being blamed for not being able to push a hoop with a stick. It's harder than it sounds. Later on in Leg 6, Nick steps back and they work together as a team to triumph over a detour that required switching, then switching BACK (from identifying music in a room full of pianos, to finding frames of film in a gigantic stack of film comprising an entire movie, then back to the pianos) Only to have to have Nick dress in a dress and shove cow manure on potatoes. After the detour, 6th place had finished and they ran over to the mat in last. It was a non-elimination leg.
Also in Season 17, one team was comprised of a mother and daughter, who had pretty much never met.
Yet again in Season 17, Chad and Stephanie were a team of a dating couple, who had quite a bit of friction in the race, but learned in Moscow to start trusting each other, and they became much friendlier to each other. The next leg, in Oman, Chad decided to propose to Stephanie using his mother's ring, which he had brought with him, waiting for the right moment to propose. She accepted, and their status changed from "Dating" to "Engaged". What's more they came in first on that leg, for the first time all race!
Upon winning Season 18, Kisha and Jen donated all the prize money to their mom so she could start her own business.
Even better: the episode where they won just happened to air on Mother's Day.
In Season 20, Bopper and Mark come in last, after Bopper had been talking about how he's on the Race to help his daughter, who has numerous health issues. It turns out to be a non-elimination leg, but before that's revealed, Art and JJ, who had just received $10,000 for finishing first, told him they'd share it with him.
A few times over the course of the series, a team gets trapped in a Roadblock. In spite of best efforts, the challenging member just can't seem to clear it. Multiple times. There is talk of quitting (or at least taking the 4-hour skipping penalty). There are Rousing Speeches to not give up. And then, eventually, once he/she finally clears it, the nearby natives who had watched the whole thing cheer and give what looks a lot like a group hug. That team, as expected, usually winds up coming in last, though if it turns out to be a non-elimination leg, it just makes it even more heartwarming.
James and Abba lose their money in Bangladesh, and they were forced to look for another way to pay for money. And a group of locals all donated to them and paid for cabfare.
In a later leg, the racers have to do a task that involves the locals to donate thirty Euros. An elderly lady pays ten euros to James and Jaymes to finish a task. Then when Abbie and Ryan are stuck on the same task and spend awhile asking for locals to stop (in the rain), a woman pays just enough to finish the task. Rather than cheering, all Abbie can do is hug the woman and cry into her shoulder.
Josh and Brent, who, as a gay couple, were not able to really show each other any affection in several countries during the race, finally getting a kiss in as they win the race after Josh collapses to his knees, overwhelmed.
Abbie and Ryan's (especially Ryan's) reaction to Josh and Brent winning Season 21. After spending several legs at the back of the pack with them, waiting for each other, and helping each other with challenges, they both can be seen jumping around, and cheering their hearts out as the Beekmans reached the finish line first.
Before the start of the race, Bopper was ejected and replace with Mallory to team up with Mark in Season 24 All Stars due to his pancreas. It was heartwarming that some of the teams at the airport felt sorry for Mark losing his partner even Rachel who cries at the starting point after hearing this news. This happens again when Mark and Mallory were eliminated and all the teams clap their hands for their effort in trying to win the race for Bopper.
In Season 1 of the Asian version, the non-elimination penalty required the last team to relinquish their money and weren't given any money for the next leg. What turns this into a heartwarming moment is when the next day, nearly all the other teams leave Sandy and Francesca (the last team) some money on their car under the windshield wiper. The total given was a little bit more than the teams were given during the start of the leg.
The Amazing Race Australia has quite a few of these-the most notable however is during the first leg of the second series. The winning team for that leg received the Salvage Pass, which would either give them a head start on the next leg (on top of any lead they had anyway) or save the last team from elimination. The team that won it (Ross and Tarryn) ended up using the salvage pass to save the last team (Lucy and Emilia) from elimination. Cue hugs all around upon hearing the news.
Season 2 of The Amazing Race Australia has cops Shane and Andrew shining the shoes of the locals as part of the Shine detour. One of the customers has a 3-year-old boy. Afterwards, one of the cops has a small conversation with the customer and pulls out a toy kangaroo from his backpack to give to the child. The child's face absolutely lights up upon receiving the toy, but if you look closely, you can see that the cop had actually brought along a whole bag of toy kangaroos to give out on the race.
Season 2, Leg 4 of TAR Australia had rockabilly couple Kym and Donna getting themselves lost in Dubai while trying to find the location of a racecourse (and also blowing a tyre in the process). While other teams in a similar circumstance take it out on their partner or whoever's nearby, Kym instead walks away from his fiancee to vent his frustrations and then comforts her.
Season 23's exes Tim and Marie. Marie spent an entire season fighting, screaming, and generally being a gigantic pain to both Tim and the viewers. But at the Finish Line, Marie admitted that no matter what, she really did love Tim.
Season 3 of China rush had a task where you had to eat 2.2 kilograms of an extremely spicy dish. Stars-crossed lovers Zhang Yelin and Li Yang reached a bump when Li Yang started crying from the spiciness so Zhang Yelin stopped what he was doing to play her a song on the guitar to help her feel better
Season 24's Dave and Connor. In Leg 7, they give Caroline and Jennifer second place as a thanks for helping them in the roadblock. Then in Leg 8, once Dave and Connor got U-turn, instead of U-turning the Jet and Cord, they U-Turn Leo and Jamal instead, which confuses Jet and Cord since they're near them. Also this is the first time that a team TELLS another team they've been U-turn with a good reason namely AND HELPING THEM. Leo and Jamal comment that they don't called it a U-Turn but a gift.
To top things up, Leo and Jamal even tag along with Dave and Connor once they finish before them to say thanks.