Well, you don't always have to go so far as to kill them, but yes, it's common for a character who has reached the end of their arc to get "retired". In Tony Stark's case, as somebody who started out as an incredibly self-centred weapons dealer, the end of his arc being the ultimate sacrifice to save humanity kind of works, as for Steve Rogers, the man who "can't live without war", whose entire motivation was that he wanted to join the fight, having the ending of retiring, living out a peaceful live with his love and passing on the shield to the next generation of hero.
The other solution is that a new problem could occur that builds on a flaw the character has never confronted because it was never an issue, and this is where we get Peter B Parker's arc in "Enter the Spiderverse". This is a Spider-man who has spent years embracing the motto of "with great power comes great responsibility", but then a new problem has emerged which is that he's getting older and MJ wants commitment, wants to have kids, and he just can't take on that particular responsibility, because it's an emotional one, different from the responsibility of fighting villains.
Spider-man is a solo hero who doesn't have sidekicks, which makes sense because he started out as a teenager himself, and teenage sidekicks are more something an "adult" hero has. So Peter has never been responsible for a kid; it's a new challenge he's never solved and his response was to run away from it, causing MJ to break up with him. His arc in this movie is that he's put in charge of a kid and learns to stop running away from fatherhood.
So just because your character has finished one arc, it doesn't mean they can't have a new, different arc, especially if there's a timeskip and they have to deal with a new problem; like a kid character who overcomes childish selfishness then as a teenager must deal with coming to terms with romantic jealousy, or a young adult character who learned to be confident in their abilities now in middle age has to deal with their physical prowess weakening and taking on a more mentoring role.
The one big thing to avoid is constantly having the character regress. Sometimes in a bad situation a character might temporarily go backwards and fall back into old bad habits. This can work okay if done sparingly, like in say Despicable Me, where after character development, Grue faces a horrible setback and falls back into his villainous ways. If it happens too much though or every story with the character feels like a reset, or it's not obvious why the character's development went backwards, it feels like bad writing. Cloud Strife in Final Fantasy VII Advent Children has practically regressed to the person he was at the start of FFVII only less fun and sassy; he's just become this grumpy loner, and it's not even clear why. A lot of people felt like it was weird that he had to learn the same lesson of "having a support network is good" a second time.