Due to the comedic nature of the series, the characters of How I Met Your Mother wouldn't be like most of the tributes that end up in The Hunger Games. Many of the victors had some kind of training beforehand, and life in the district caused people to become much more self-sufficient than those that live in New York City.
If the gang ever got reaped and forced to compete to the death, their survival would depend on more than just their physical abilities. Their athleticism would play a role, but as fans know, winning the Hunger Games also requires tributes to think outside of the box. There are a few characters who would probably stand a chance of survival thanks to their skills and upbringing, but there are others whose morals would prevent them from making it out of the Cornucopia.
Ted Mosby
Similar to how Jonah Simms from Superstore would fend in The Hunger Games, Ted's morals would be his downfall. It's been mentioned several times in the series that Ted is against guns, and it's hard to imagine that he'd be fond of the other weapons the Capitol provides the tributes. Ted's not the best at reading the room, and although he can be loveable, his head is stuck in the clouds.
He believes that the universe has a plan and everything is going to work out the way that it's meant to. He would probably try to organize secret meetings and form a peaceful protest against the Capitol while everyone else is off training. Rather than grabbing a weapon Ted would be trying to unite everyone in the Cornucopia and would end up dead within the first few minutes.
Tracy McConnell
Like Ted, Tracy would have a hard time hurting anyone who wasn't posing an immediate threat. She'd find all the children who needed help and get them to safe a place. Tracy has never been shown to have any training when it comes to self-defense, but she can certainly fend for herself in other ways.
Although telling someone off isn't the same as killing them, Tracy exemplified with Darren that she can stand up for herself if the situation calls for it. But if it was down to the wire and the remaining tributes were worth her sacrifice, she would likely give her life for them.
Barney Stinson
Barney isn't the most courageous character when it comes to physical confrontations, so there's no way he would be prepared to charge headfirst into danger. His morals wouldn't be a problem since Barney is usually out for himself, but as shown in multiple episodes, Barney can't fend for himself in a fight.
Laser tag isn't very similar to the threats Barney would have to face in the arena, but it's better than nothing. The owner himself said that Barney might be the best laser tag player he had ever seen, and Barney could probably use it to his advantage. He knows how to avoid getting shot by a laser, so hopefully, the same tactics could apply when being shot at by an arrow.
Lily Aldrin
Lily has two completely different sides to her personality. As a teacher, she is quite literally all sunshine and rainbows when it comes to her students. Lily doesn't condone any kind of bad behavior from them and teaches the kids that the most important thing is having fun. Therefore, at school, she doesn't exemplify a very competitive nature.
But outside of her job Lily has no problem letting out a different side. She can be cutthroat when she wants to be and when she gets angry, her "dead to me" stare is rumored to be downright terrifying. Lily won't back down from a fight, and even if she's not athletic, she could use her intelligence to her advantage.
Marshall Eriksen
Marshall's kindness might make him appear as less of a threat, giving him the upper hand he needs over his opponents. His good nature and desire to help others would make it difficult for him to participate in the Hunger Games, but he's proved over the series that he will resort to violence in necessary situations. He even wins a slap bet with Barney and makes sure that every slap is more painful than the last.
When Ted and Barney are in danger, Marshall comes to the rescue, revealing that the fights he used to get into with his brothers were pretty hardcore. He'd be a tough competitor on the physical side of things, and his background in environmental law would likely help him adapt to his surroundings.
Robin Scherbatsky
There's no doubt that Robin Scherbatsky would have the greatest chance of winning The Hunger Games. Her father trained her to be prepared for any and all situations, and he even dropped her out of a helicopter and into the wilderness as a teenager.
Robin is the toughest both physically and mentally, and it would take someone very lethal to get one over on her. She's an excellent shot, so it feels safe to say that she would have no problem with using whatever weapons the Capitol provided.