A bean bag chair is fine until a seam gives out, and then it is thousands of tiny beads everywhere at once. Double-bagging it before the move is cheap insurance against that exact scenario.
Pro tip: If you have a vacuum storage bag on hand, use it. Compressing the chair for the move makes it dramatically easier to carry and fit into a packed truck.
Watch out: A burst bean bag seam releases thousands of small beads almost instantly, and foam beads are a serious choking and inhalation hazard for young kids and pets; double-bag before moving it.