As far as whether or not any account data is stored, it really depends on the implementation: using OpenID for authentication doesn't automatically guarantee that the account data is not stored in the forum's database; there may be no password data (or it may be NULL), but there's likely a username in the database to bind to the OpenID userid.
Since alot of these forums were developed before OpenID became popular, there's likely legacy data structures and/or hooks into the local user data, so while authenticating is done by a 3rd-party, some of that 3rd-party data may be replicated locally in the forum database by default; if anything, the "forum-centric" metadata like the user's threads, posts, and maybe user panel stuff like homepage, city, etc. would most likely be stored locally.
Having said all that, it may not matter to you, but if it was for strict privacy reasons, you might not be able to pull it off without some modifications.
As for options, PHPBB3 has a 3rd-party add-on to allow OpenID authentication. There's also Simple Machines forum and OpenID is supported in the 2.0 Release Candidate 4.