I imagine it’s possible with a plugin.
This would be pretty close.
https://bbpress.org/plugins/topic/bb-moderation-hold/#post-245
It holds the post for moderation. You could probably modify it to not hold for moderation, or if nothing else use the same filters and actions for your own plugin to just email on a new post or topic.
I would register the New Posts RSS feed with FeedBurner and subscribe from there.
Oh yeah, RSS feeds are built in 
Good idea.
This is very dumb of me, but I can’t find anywhere under the bonnet to register New Posts RSS feed with Feedburner (it’s a while since I set up my WordPress blog, but I seem to recall in WordPress, there is a place to do this).
I guess I go to Feedburner to register, then? Once I’ve done this, does this mean that every post will be notified to anyone who signs up for the feed?
Yes, I think it’s called “Burn a feed” at Feedburner.com. You can do that, then either log in or create an account. Once that is done, you will have a URL to insert into your bbPress header ( I think ) to redirect your feed to the FeedBurner feed.
Then, anyone who subscribes to that feed will be notified of new posts (if you burned the feed for Recent Posts, not Recent Topics.)
I’m not sure why it needs to go through FeedBurner though. If you have an RSS feed already in your bbPress forum for New Posts, why not just subscribe to that to be notified of new posts? I must be missing something…
P.S. There is nothing built into bbPress like this, for FeedBurner. So, you didn’t overlook it: it’s not there. Do the work at feedburner.com then change the URL for the feed in bbPress.
an email for every new post would be easy to do but wow your inbox could go crazy after awhile – not to mention your server sending tons of email might get it flagged for spam
