If my game closes before I open a gift box I received from playing regular scenes, will the gift box..

If you mean the the pink and blue star boxes you get when you fill the stars in the regular game, they will accumulate. You can find them by tapping on the blue satchel next to the shop icons and open them at your convenience. Many people will save them until they need them.

If you mean the suitcases that you receive randomly at the end of the game, their contents will immediately go into your game account, before you see them open.