Harry Potter
Sometimes recasting isn't a decision so much as just a fact of life. Richard Harris passed away in 2002 as a result of his Hidgkins Disease. Though he was acclaimed in his role, earning two Oscar's for it, the show must go on. He was replaced with the actor Michael Gambon, who appeared in the remaining six films.
Friends
The core friends in Friends stayed the same throughout the series, but the periphery characters had their fair share of recasts. One of these was Carol, Ross's first wife. Anita Barone originally played Carol Willick, but she was replaced in later seasons with Jane Sibbett. The first actress was offered a full-time gig on another show and took that instead.