flooBB is an easy to use bulletin board system which does not rely on an external database. It simply uses text files to store the data and is optimized for speed. It incorporates TinyMCE to provide a WYSIWYG method of entering posts into the system.
A PHP 4 Open Source Flat File Based Bulletin Board System -> GrimBB uses text files to store the data for posts (doesn't need outside database like MySQL). Its easy to configure (only two distribution files require changes to achieve immediate results).
Navboard is a php based web forum using text files to store data instead of using a database, compatible with any server running php 4.0 or higher, all the features of most major forum software and no installation needed.
XMLGroups is a bundle of scripts that enables you to display/manage newsgroups.
All data of this newsgroup is stored in an XML file.
The difference with tradional forums is that it is very easy to set it up. You don't need to have access to a database