Update 7: I managed to track down the cause of the 403 errors some people were seeing and I do believe it's been fixed (or at least that's the hope). I know at least one post that was causing issues has now made it through, so hopefully the rest will work.
As for the cause, well it looks like we ran afoul of an updated security patch on the server. It was a bit to stringent and blocked requests that it thought was questionable and apparently we qualify.