Does your monitor have integrated speakers? Does the Sound card recognise a HDMI connection from the GFX card?
It's almost like the HDMI port is disabled on the GFX card...
Have you tried or checked the following:
Is the HDMI lead fully inserted in the socket? I know if the GFX card is installed a little skewed the HDMI port on those cards is right next to the PCI slot dividers on the chassis and this can prevent the HDMI lead from being fully inserted into the socket (this happens a lot actually and is easily overlooked)
Do you have a second screen to plug in to the DVI? In theory, you should be able to connect both the HDMI and DVI leads to the same monitor and switch between them (well I can on my BenQ screens anyway)
Can you see if it will pick up a TV with the HDMI connection?
Do you have another HDMI lead to try?
Is there any chance you may have shorted the HDMI lead while connecting things up and you've buggered the HDMI socket on the GFX card (you'd have to be one unlucky SOB for that to have happened though

)