It isn't possible to customize the New Posts feature in an officially sanctioned manner (I believe a third party hack might do this, but our tech team is unwilling to install it for various reasons).
However, one can customize the display of forums one subscribes to. So instead of filtering out forums, you are seeing only forums you subscribe to. You would view that list in your User CP. See below for instructions.
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How To View Only Those Forums You're Interested In
Subscribe to those Forums you're interested in by clicking on the Forums Tools link and choose the notify in CP (control panel) option. Then when you visit the site, go to your Settings. There you will see only those forums you have subscribed to. Forums with new content will be indicated by the standard iconography. There will also be a section of New Posts in Subscribed Forums.
Then in each forum you've
subscribe to, change the Thread Display Options to Sort Thread By Last Post Time to ensure the newest threads appear at the top.
Set a link in your browser to your Settings page and you now have any easy way