It is not possible to replace or remove file attachments when editing a discussion post or reply. However, I have found a way to accomplish the same thing.
Every file attachment is added as a Library entry. The author can edit or delete file attachments from the Library view.
There is no straightforward way to find the exact library entry that corresponds to a file attachment on a discussion post or reply. You just have to search for it and click through the pages of results until you find the right one. However, once you do find it, if you are the original author, you should be able to Edit or Delete the attachment from the Actions menu.
It would be nice if Higher Logic did not make the process so needlessly obtuse, but at least there's a way.