How does the googlebot see a phpbb forum page? i.e. will such a page
be ranked, or penalized, or ignored?

little jew

Googlebot uses a database containing most websites in the world. It
doesn't find a new website (or a forum page) by itselves. It finds
your forum when it is linked on a page it already knows or if a user
with google toolbar visits this page (depends on the settings of the
toolbar). If Googlebot has found your forum it will follow all other
pages linked on the main page and even the links on the linked
pages....
The more links to your page it finds, the higher is the ranking for your site.
So there's nothing special with a forum page but that there are
dynamical links that more and more spam the google database.

Some forums use so-called "session ids" which are almost totally ignored by google:
://www.google.com/webmasters/2.html#A1
"we limit the amount of dynamic pages we index"

Nevertheless, some forums appear to be static sites since they
automatically recognize googlebot and don't use session ids in this
case.
If the front page of your forum has been indexed and you don't use
those dynamic links, google will index the whole forum.
Concerning phpBB, there's an easy work-around to make google spider the subpages:
http://phpbb.com/phpBB/viewtopic.php?t=32328&start=0&postdays=0&postorder=asc&highlight=short+url