Yes, it is possible. Often, that's called "passive ownership." Owning a significant share doesn't necessarily make one part of the company management.

No, they don't have to sell. Many companies have what are referred to as absentee owners. The role they do play is selecting the management team

Say someone inherits controlling interest of a company, but doesn't want to take on the responsibilities that come with being a major shareholder.

Is it at all possible for them to retain these shares without playing a role in the company? Or would they have to sell their ownership?