You need to unsubscribe from the forum.
Hope this helps,
NB - Whoever set the forum up **may** have set it up so you cannot unsubscribe. If that's the case you need to speak with them.
I don't recommend "unsubscribing" unless you are completely finished with a forum.
I have not observed the problem that you describe. What I do observe is that sometimes Moodle will not say I have new posts, and I do. Even with forum tracking enabled.
In your profile, have you looked at your setting for "Forum Tracking?" Try changing this.
Also, some of this depends upon the specific forum, and its settings about tracking.
Could you share a screen shot of your problem?
If you are talking about the notifications on the MyMoodle page that drop down below the course title. Logging out and logging back in after reading all posts will refresh that notification.