Sounds like a driver issue. I'd try uninstalling the AMD drivers (in safe mode, if necessary), rebooting, and installing the latest AMD chipset & "Omega" display drivers off the AMD website. Before reinstalling, I'd also note whether the HDMI works with the drivers uninstalled; if so, that would confirm the problem to be driver-related.
If you're using a laptop, another thing to try would be to press Win-P (hold down Windows key and press P) to bring up Projector mode options, which allows you to select which output, or both, to use.
You also didn't mention which OS your system's running or if you're running a desktop or laptop; you should either be running Windows 7 SP1 or Windows 8.1, fully patched. If you have 8.0, install the 8.1 update; if you have 10.0 Technical Preview uninstall it. Win10 has several video driver issues, especially with the latest build.