The Prince Charles, Prince of Wales will be her and Prince Philip's successor.
When Philip dies, Charles will be Duke of Edinburgh
Actually from what I've read the dukedom is being intended for the Queen's youngest son, the Earl of Wessex (yes, unlike his bros he's a mere Earl. Sad!) A reigning monarch doesn't hold any peerages by custom and convention, so Charles can't succeed Philip as the Duke of Edinburgh, and Andrew is already the Duke of York.
So does Harry become Duke of York in addition to Sussex when Andrew dies?