This issue occurred due to the default web site having improperly configured WebDAV handler mappings and modules. A new web site was created in IIS. Under that, a new web application was created for Forms with the physical path set to a copy of the Forms folder. Using this new site and web application, Forms functioned normally.

A general tip for troubleshooting Forms would be to always check the application event log on the Forms server for details. If the event log doesn't have anything, then use a browser like Chrome and bring up the element inspector. Then click on "Network" and recreate the issue. This will help troubleshoot errors that occur if the web request can't even make it to the Forms API.