I have a book in iBookstore. (Not made with iBooks Author, but I couldn't find a likely forum for ePub).
I want to move a chapter to a different location (in the various appropriate files: toc.ncx, content.opf, and toc.xhtml), export, and upload the new ePub version to iBookstore. Content isn't changed, just the location.
Would this change require a new ISBN number?
No, you don't need to have a new ISBN. You can make far more extensive changes without also changing ISBN, including replacing an entire chapter with completely different content. (As I have done with a printed book that replaced an entire chapter for the third printing, due to changes in the standards landscape.)
You should, however, identify the "printing" of the book in the front matter, so there is some way to distinguish versions.
In general, when to assign a new ISBN is entirely up to you. As long as you still consider the book to be substantially the same after the changes, the old ISBN is fine.