I figured it out and thought I would share for the benefit of the community. If you want to make it so your 'New Thread' button and links do not appear in forums where users do not have permission to post New Threads, you need to edit your 'FORUMDISPLAY' template. Find each instance of this:
PHP Code:
<if condition="$show['newthreadlink']">
and replace it with this:
PHP Code:
<if condition="$show['newthreadlink'] AND ($forumperms & $vbulletin->bf_ugp_forumpermissions['canpostnew'])">
There should be three instances. One for each button, top and bottom, and one for the drop list link.