Figured it out! In my theme's .tpl file, it had custom HTML inside of <div id="Foot"></div>, but no {asset name="Foot"}. After adding {asset name="Foot"} in between the "Foot" div, everything seems to be working perfectly now (knock on wood).
When you're in BBCode mode and are starting a new thread, examine your page using FireBug or Chrome's Toollbar. Look at the Foot element of the page, found here: body #Frame #Foot. Inside that you should find the #Definitions at div #Definitions. What does the "InputFormat" definition contain?