This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Hell's Kitchen
Season 3, Episode 9. The episode before, Julia was eliminated and Ramsay had promised her she's off to culinary school. What's going on with everyone else? Bonnie is heartbroken that she had to put up her close friend to be eliminated and is hugging Jen. Rock? He's sitting alone, hitting his head with a fist in dismay because Julia was such a Worthy Opponent to him.
The end of that episode is also heartbreaking as Ramsay sadly stated that Julia was eliminated and how he promised to send her to culinary school, hoping that she would come back one day to win the competition hands down.
Robert withdrawing from the competition in Season 5 due to health problems. As one of the most passionate chefs that season, he worked so hard and fought his way to the final 5, only for him to get his dream taken from him by a heart condition. Right in the middle of seeing the prize restaurant. To add to it, Robert cancelled his wedding just to compete.
Not to mention Ji's withdrawal in the second episode of that season due to an accident where she sprained her ankle. Her teammates were all crying (and some of the men even appeared to be on the verge of it) because they felt she was a strong competitor even with her injury.
Season 5's Danny may have come across as a bit arrogant at times. His final confessional, after winning the competition, shows him crying, knowing that his late mother is smiling down being proud of him.
Nilka's ejection as a whole, in season 7. After repeatedly messing up, Ramsay is finally fed up enough to force her out, but she refuses to give up to the point that she becomes a detriment to the work of the remaining team. Compounded by Ramsay's tonal reversal when he personally wishes her well as she truly leaves the studio.
Season 9's Paul revealing that he was on Hell's Kitchen for his mother, who apparently died not long before he was on the show. Watch the finale, and try not to cry when he wins.
Season 11 Finale. When it was revealed that Ja'Nel was the winner, Mary is crying because while she wants to be happy for Ja'Nel's success, it's a bitter pill to swallow. To twist this moment even further, Dan (who was acting like a total jerk throughout prep and who slowed down dinner service to the point that Mary had to kick him out) goes to Mary and hugs her, apologizes for his behavior and promises to make it up to her someday. Question, Dan: Where was that kind side of you throughout the season?!
In a way, Simone's leaving Hell's Kitchen at the end of Season 12, Episode 3. Whether she was faking illness or not is still up to speculation, but she sounds so defeated and just says "It's ok Chef, I'll leave" instead of giving Chef Ramsay a chance to announce who is leaving.
Anything involving Aaron Song since his death in 2010, especially his brief appearance in season 4.
Two people from Season Two are no longer with us. Rachel shot herself in 2007, and Keith drowned in 2012.
Even worse if you remember how close they were to Heather. She must have been devastated.
Just recently, Paulie from Season 16 passed away from an accidental drug overdose. Even if people didn't like him, his death is still very tragic.
Jessica's elimination in season 12, she has a complete mental breakdown and begs not be kicked-off the show, even with her mistakes it's really hard to watch and not feel sorry for her.
Ashley's elimination in Season 13. Throughout the season, she remained blank-faced and emotionless, but in the interview afterwards, she's seen crying, saying all she wanted was to work with Chef Ramsay and that it really hurts knowing he thought she wasn't good enough. It's made worse since she arguably didn't deserve to be eliminated (Roe and Steve were up for the second time in a row).
Season 13, episode 9: Steve is forced out of the competition due to a leg injury. Ramsay even expressed disappointment, as he felt that Steve had been a solid competitor throughout the season. What makes it worse is that Steve clearly wanted to go on, but Chef Ramsay said he couldn't let Steve sacrifice his health for the show.
Frank's elimination, too, for that matter. He takes it gracefully, but is visibly holding back tears and is very upset with himself. It's made even worse since he wrote Chef Ramsay a thank-you note earlier that episode for not eliminating him the previous night (which is sort of happy-tear-jerking, since Ramsay expresses gratitude for it).
In Season 14, Bret suffered a fate almost identical to Steve's the previous season, being removed from the competition due to a slipped disc in his back by Chef Ramsay, despite being considered one of the best (if not THE best) chefs on the blue team.