Better yet: 301 to the correct page.
Edit: since this forum is gone, 301 to a page that explains that the forum has been permanently shut down. That's because external links still point to interior pages of that dead forum. That's not your fault, but on the other hand you sure enjoyed collecting all that free link juice. Surely this has happened to you: you click a link that seems perfect, only to land at some jerk's homepage and go "WTH, this stinks." Back up, look at the URLs and find out the jerk destroyed the content you needed. And in that moment, who are you angry at? Bingo.
As far as Google mistaking that for doorways, no worries - people shut down forums all the time, and that's certainly better than leaving them up to be stuffed full of spam by robots. I don't know if your platform allows this, but the HTTP protocol allows you to serve a 404 that is both a redirect and an actual web page: include a Location: in the HTTP header, and an HTML header/body with the usual redirects.
The crawling continues because external sites still link to the old page location; or Google is aware of that page being bookmarked by users. That is absolutely free link juice that you have earned! If it brings enough traffic to be worth the bother, and the content is any good, restore the forum in archive mode.
Even if you 410, that doesn't mean Google will disappear from the logs forever; as long as those external links and bookmarks live, the Free Traffic Fairy (er, Google) will check up periodically to see if the URLs have Lazarus'd.
As far as the access log pollution, grep -v ’/forum-URL-pattern-here/.*404' Though you may want to know about organic traffic to those pages; it is worth money.