It tends to go through stages.
First stage: you might see a sudden bump in new members as you publicize the new forum.
But because there aren't a lot of members or posts, there tends to be a lag in posting at first. Seeding new threads with posts that encourage others to post their opinions helps. Contests can help.
Second stage: gradual increase in registrations and a slow increase in posts. This may continue for weeks or months. depending on the niche (focus of your new forum).
Third stage (if you last this long): Your forum reaches a "critical mass" of members and threads/posts and the growth of your forum starts to accelerate significantly. You are now established in your niche.
But even after you hit stage three, a forum always depends on the active participation of the administrator / owner. If you slack off, so will your membership.