I think they've been largely replaced by reddit. Autonomous subreddits aggregated into a single feed instead of separate global subforums, upvotes instead of bumps (that also serve to self moderate), threaded conversation instead of quote trees, reddit is basically just forums adapted for scale.
And I would argue that that is why Reddit sucks so much (personal opinion - I can understand why someone might enjoy the community there, but I do not in the slightest).
Don't get me wrong, I enjoy the news aggregation feature, but the community aspects are terrible. The benefit of forums was primarily in their segregation of a small-but-active community. Reddit works in precisely the opposite manner, so you end up with boring, repetitive, least-common-denominator tripe. Any subreddit small enough to avoid that is too small to be worth the time - it seems that small communities (by which I mean actual communities, as opposed to random conglomerations of people with vaguely similar tastes) tend to have other sites and methods of communication they frequent, such as this one.
I think that the poor scaling of forums is their greatest strength, and Reddit is an example of what happens when forums do scale.
