To my knowledge, no current forum software solutions have this, for several reasons, some of which I'll list here:
1) Authentication. It's all too easy to fake an email authentication token and have spam posted to the forum
2) Use of features. What's the point in spending money on a forum solution that's had loads of work on rich user experience, user interface and interactive features when you delegate all of that experience to an email client?
3) Advertising. Lots of forums make money by running AdSense or a similar ad scheme on their pages. With email interaction, there is no need to visit the forum at all, resulting in zero advertising revenue.
4) Choice of medium. Forums are web-based. There is an email-based medium, and it's called listserv (or various other email-based communication methods). If you really want an email-based solution, why opt for a forum?