Just watching House of Cards season 5, Claire becomes acting president elect due to neither Underwood or Conway getting the required electoral votes. Thus the vote for President is left to Congress. Given that it is entirely possible that Congress elects Conway, could Conway get rid of Claire if he won? I believe the term they used in the show was a split-election. A reporter also was asking Conway what would happen in such a case, and if the 12th amendment prevented him from getting rid of her if he won.
From what I can gather, the VP is the only person in the President's staff that can't be fired by the President.