The New Posts icon, is not a "posts not read" icon. It is an indicator for forums and threads that have posts in them since your last visit. "Last visit" is determined by a period of inactivity. The length of the inactivity is determined by the Reset last visit: option in the General Preferences section of your preferences page.
Here is the explanation provide on your preferences page.
The forums keeps track of when you visit and when you leave. After you have been inactive on the forums for a certain amount of time your last visit time will be reset. This is what determines whether new post icons are displayed. By default the forums resets your last visit time after 30 minutes of inactivity.
If you check the forums off and on over a 2 hour period you might want to have the time extended so that you do not miss new post indicators that would normally disappear after you've been inactive for 30 minutes. Choose one of the other time options to extend this reset time.
You can turn the icon(s) off by clicking on the Mark As Read link on each sub-forum's topics page or by clicking on the Mark topics in all forums as read link on your My Forums page.