Ummm... well here's the thing. A forum itself cannot exist without a database, and an sql language. I'd assume you would use MySQL.
Now, all database scripts would be written in a scripting language such as php, because flash itself cannot directly communicate with databases (not to my knowledge) but flash can call php files and pass variables to them
So in the end flash would just act as an interface between the database and the scripts that modify it. So basically the most you could have would be a "post your message here" screen done in flash. Flash would send the variables to a php file, and then the php file would update the database with the new post.
Not really worth it if you ask me, but meh