This is a known issue with org-mode and fci-mode. See here and here for a discussion and some suggested solutions. Personally, I found the cure to be worse than the symptom and it didn't quite fit with what I wanted.
I have instead found an alternative which works with org-mode and still provides me with the sort of information I was after with fci-mode. I found it was trivial to get similar functionality with whitespace-mode. Rather than a vertical line indicating the fill column, a face is used to colour all characters past the fill line with a different colour.
(setq whitespace-cleanup-mode-only-if-initially-clean nil
whitespace-style '(face lines-tail))