No, they can't share your close friends story directly. The whole point of a close friends story is that it's meant to be more private and only viewable by the people you've selected as close friends. Tagging someone in it doesn't give them the ability to share it outside of that close friends group.
They cannot share it. Close friends stories are for a limited and private audience. When you tag someone in it, it's just to draw their attention within the context of that private story. Instagram doesn't allow sharing of close friends stories by those who are tagged in it or even by the original creator in a way that makes it public or accessible to a wider audience.
Nope. The nature of a close friends story is privacy. You choose a specific group of people to be your close friends and that story is only for them. Tagging someone in it is just to engage them within that private circle. If they could share it, it would defeat the purpose of having a close friends - only story. So, sharing is not an option for someone who is tagged in your close friends story.