Valhalla, this depends primarily on the position of the chimney breast relative to adjoining houses, age of the house etc.
If it adjoins another house, it could be tricky and would advise getting a structural engineer to look at it. If it doesn't, (ie if you house is detached or your chimney is not on the party wall) it is very esay to remove. I would advise however to remove it all completely (ie from the roof down) and fixing the roof. Ceiling joists and floor joists would normally be trimmed around the stack and not bearing on them so they wouldn't be a problem.