A few times over the course of the series, a team gets trapped in a Roadblock. In spite of their 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 been watching 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.
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 a liability for Zach, especially in the later half of the season, these words were especially heart wrenching.
Similarly, Ian was seen as somewhat of a blowhard. 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.
At the Leg 5 Detour in Portugal, Michael decided to face his fears and do "Ropes" (rappel down the cliff) instead of "Slopes" (walk down the hill). When Flo had trouble on the same task (before switching to Slopes), he tried to reassure her that it was okay and that he was scared too. Many contestants have been less supportive than that toward their own partners, let alone someone on a rival team.
In Season 4, Reichen and Chip's Fast Forward meeting the world's most adorable orang-utans.
In Season 5, coming in last, Charla and Mirna end their race by thanking Phil for the experience, and Charla (who has dwarfism) 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.
Kris & Jon's relationship.
In Season 7, Uchenna going back to help Meredith and Gretchen with "Brawn" even though they were racing for last place.
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. After knowing that they won't win the race and just wanted to move on, 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.
One team was comprised of a mother and daughter who had pretty much never met. While race journey was cut short, they had a more soul searching relationship in Elimination Station.
Chad and Stephanie were 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 and got them a honeymoon trip!
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.
Earlier in that same season, the teams had a long wait for a train that would take them to their next destination. One of them found a basketball court, and one of the Globetrotters had a basketball. Zev and Justin, Flight Time and Big Easy, and Keisha and Jen had a 3-on-3 game, with Jamie and Kara cheerleading, and everyone else just enjoying themselves watching. After all that they went through and the stresses of the race, it was nice to see all the teams come together and blow off some steam just enjoying an activity and having a good time together.
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.
Season 21, 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. 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.
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.
Before the start of the race, Bopper was ejected and replace with Mallory to team up with Mark in Season 24 due to his pancreas. It was sad that some of the teams at the airport felt sorry for Mark losing his partner. 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.
Leg 4, Brendon & Rachel and Margie & Luke were struggling at the Cocktail Detour and were fighting to avoid elimination. When Luke just starting to have a tantrum, Rachel gave him a pep talk so that he shouldn't give up. While this decision turned out to be horrible for Brenchel, Margie & Luke were grateful for her and admit that they saw her in a positive light. Fortunately for Brenchel, it's a non-elimination leg.
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, they U-Turn Leo and Jamal. When both teams met, Dave & Connor TELLS Leo & Jamal they've been U-turn with a good reason namely AND helped them in the Detour. 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.
Season 30's team Ocean Rescue has been dating for 9 years at the time of the race, and Lucas secretly wants to find a perfect moment to propose... but they have a weak showing in the first three legs, and Lucas and Brittany are having a really bad go of it in Leg 4. They are barely spared elimination and got into a bad argument, and conversations indicate Brittany's father doubt's Lucas's commitment to his daughter. Then, in Leg 5, they spring ahead at the Roadblock and stay in the lead to everyone's surprise ("We're being beaten by Ocean Rescue..."), finish the Detour, and get to the Pit Stop first, trailed by the X-Games girls Kristi and Jen. The camera focuses on Lucas's hand slipping into his pocket... and then after a rather obnoxiously-placed commercial break, we see Lucas propose to Brittany and slide a ring on her finger. Several more teams arrive and celebrate their engagement and we learn Lucas and Brittany have resolved to stay together no matter what and the arguing made Lucas realize how important it is to keep by his fiance.
Cody and Jessica deciding to tie the knot after winning The Amazing Race 30.
In Season 1 of the Asian version, the non-elimination penalty required the last team to relinquish all of their money and not receive any money for the next leg. The Heartwarming Moment comes on the next day, when nearly all the other teams leave Sandy & 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.
TAR Australia Season 2, Leg 1: 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 & Tarryn) ended up using the Salvage Pass to save the last team (Lucy & Emilia) from elimination. Cue hugs all around upon hearing the news.
Leg 4: Couple Kym & Donna got themselves lost in Dubai while trying to find the location of a racecourse and also blew a tire in the process. While other teams in similar circumstances take it out on their partner or whoever's nearby, Kym instead walks away from his fiancée to vent his frustrations and then comforts her.
Cops Shane & Andrew shining the shoes of the locals as part of the Shine detour. One of the customers has a 3-year-old boy with him. 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 during the race.
They also gave a toy to the boy they had to teach 10 slang words to.
Season 3 of China Rush had a task where you had to eat 2.2 kilograms of an extremely spicy dish. Star-Crossed Lovers Zhang Yelin and Li Yang reached a obstacle 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.
China Season 2, Leg 2 had a Roadblock which involved bungee jumping, which the racers' pre-race interviews showed was a challenge they would absolutely fear facing. All of the female racers visibly demonstrated their fear of heights to varying degrees, such as Wu Xin and Baoyi breaking down into tears en route to the top. In the end, they managed to do the Roadblock with one another's motivation and comfort and all took the plunge successfully.