I was getting a 403 error when attempting to post to my blog. Found a whole bunch of info about "curl" and permissions issues, post length, etc. In my case it was none of the above, so I'm posting this FYI for those concerned.
I removed a long Amazon associates link from the post and the Save Draft worked fine. Put the link back in and got the 403 error. It was a link to a search result, with two specific key phrases separated by a "pipe" like this |. I examined the link, and decided to remove the second key phrase and pipe. Voila.
So, if you're getting a 403 error on post, check your links, it could be a delimiting character or something throwing off the WP script.
Also, if you're copy/pasting from a text editor, make sure you are working in a plain text format, not rich text. Hidden formatting characters can screw things up too.
Hope this helps.