farscapI have a standard Dell display hooked up to the DVI port. It works just fine.  To the HDMI port I have connected a Sony Bravia 40 inch flat screen television.

I think a lot of these types of cards will allow only 1 digital connection and 1 analog connection. For the digital connection, you must choose DVI or HDMI, not both at the same time. I can't find any real specifications on the web, but I'm pretty sure this is the case.
 
Try hooking up your Dell display with a VGA cable, and see if your HDMI to your TV suddenly works. You may just want to turn off your computer, unplug the Dell screen entirely, and boot up with just the TV attached to the HDMI port only.
 
I'm not really sure why they sell these really old DX 10 cards anymore, as you can get a GT 610 for the same price or a GT 620 for maybe 15 bucks more. The 8400 GS was released in 2007 (7 years ago, which is ancient in computer terms, especially when buying a "new" card). I realize you're not trying to play games, but still. To be honest, I was surprised that it is listed as HDCP-compliant.
http://en.wikipedia.org/wiki/GeForce_8_series#GeForce_8300_and_8400_Series
 
Regardless, try using VGA + HDMI and see if it works.

Sprinx

farscapI have a standard Dell display hooked up to the DVI port. It works just fine.  To the HDMI port I have connected a Sony Bravia 40 inch flat screen television.

I think a lot of these types of cards will allow only 1 digital connection and 1 analog connection. For the digital connection, you must choose DVI or HDMI, not both at the same time. I can't find any real specifications on the web, but I'm pretty sure this is the case.
 
Try hooking up your Dell display with a VGA cable, and see if your HDMI to your TV suddenly works. You may just want to turn off your computer, unplug the Dell screen entirely, and boot up with just the TV attached to the HDMI port only.
 
I'm not really sure why they sell these really old DX 10 cards anymore, as you can get a GT 610 for the same price or a GT 620 for maybe 15 bucks more. The 8400 GS was released in 2007 (7 years ago, which is ancient in computer terms, especially when buying a "new" card). I realize you're not trying to play games, but still. To be honest, I was surprised that it is listed as HDCP-compliant.

 
Regardless, try using VGA + HDMI and see if it works.

Thanks.  Had I known the card was so old, I would have never purchased it in the first place.
 

 
GT620 for $55.  I am assuming it has the capability of driving both DVI ports at the same time?  Or one DVI and the mini HDMI?
 
I will try your suggestion of using the VGA port instead of the DVI port.  I had not considered that option.
 
It could also be the television, like ErinW suggested.  Maybe I will try running the VGA from the computer to the tv's VGA input.  Though I wouldn't get any sound, which is a big drawback.