Both title matches for Aoki and Kimura could be considered this, much more the latter since the end result prompted Kimura to retire.
Kimura's title match was already bittersweet as it is. Kimura loses? He retires. Mashiba loses? Hints of more struggles for both him and Kumi await.
A good chunk of the fighters' backstories.
Ippo: Father died during a fishing expedition, had to help his mom run the fishing boat as a child.
Sendo: Mother died during childbirth, father sacrificing himself while doing his job as a fireman.
Miyata: Witnessed his father get utterly destroyed by Randy Boy Sr., watching him turn to alcohol and seeing his mom leave the family, while still wanting to follow in his dad's footsteps as a counter expert.
Mashiba: Parents died in car crash, had to raise himself and Kumi while still only a teenager.
Sawamura: Father died, mother got into an abusive relationship, which abruptly ended with Sawamura stabbing him. He was abandoned shortly thereafter.
Volg: No news about his father, his mother winds up being really sick so he takes up pro boxing to help her get better. Made even worse when we find out he fails.
Ryuhei Sawamura: Retired as the Japanese Jr. Lightweight Champion.
More specifically, it's been mentioned at various times before, during and after the match, that Sawamura was earnestly trying to win the belt. Despite all that cheating and injuries sustained, he got what he wanted... only to get into a motorcycle accident that leaves him unable to fight in the ring anymore.
Another heart wrenching moment is After Sendo has defeated Volg for the championship, Volg's contract with his gym had been severed, and could have been severed any time he lost too many matches. Volg's farewell to Ippo at the airport, knowing that they might never be able to meet again, is incredibly heart breaking. Specially when Ippo can hold back the tears when in front of Volg... but he breaks down crying once he's gone.
Watching Takamura of all people bawl his eyes out when he finds out the bear he spared (because she had cubs) got shot and turned into his stew is pretty hard to watch. Mostly because Takamura can apparently be quite motivational when he's feeling strongly about something.
Eiji Date's rematch against Ricardo Martinez. An accidentally made Game-Breaking Injury is what lead Date to not only lose against Ricardo but also retire from boxing permanently.
How about Saeki's match against Itagaki? Seeing him lose like that is heartbreaking, because he was fighting for Karasawa, who retired but was unhappy with his life. This is likely something that happens among boxers in real life, which is harsh.
The final episode credits of Rising dedicate the whole season to Kenji Utsumi and Ichirō Nagai, who are the respective voice actors for Kamogawa and Nekota. Utsumi passed away in June 2013, therefore having to cast another VA for Kamogawa, and Nagai in January 2014.