Sorry my post was so vague, I literally had no idea what information would be useful to troubleshoot this. 🙂
When: I'm not sure, somewhere between a few days and a week.
I didn't install a firewall extension, and I do use Akeeba Admin Tools which has always co-existed happily with BP. I haven't changed anything on the server, but that's not to say my host hasn't tweaked a setting somewhere.
Come to think of it, the only new extension I installed recently (and then subsequently uninstalled) was AutoTweetNG - but again I can't be sure that this coincided with BP throwing the 403 error.
EDIT: I just tried again now and it works with published articles, but throws a 403 when I try to preview an unpublished article.
I think the first thing to do here is ask your host if they can see what id blocking those urls (triggered by Better Preview) from loading.
It is probably down to some security module installed by the host on the server.