I found the answer at Impossible as Possible. This solution saved me hours of digging.
SharePoint 2010 has an enhanced security model. One aspect of this is that the setting for browser file handling is set to strict by default. As explained by Microsoft here, this is by design to prevent malicious scripts from exploiting security.
To resolve the issue simply open Central Administration\Application Management\Manage Web Applications. Scroll down until you see "Browser File handling". Change it from Strict to Permissive as shown below.
You can now view html files in the iFrame as desired.