Can you provide a copy of your Dxdiag please? (Start -> Run -> dxdiag -> Save All Information). Additionally, can you provide a copy of your Intel® Graphics report? (Ctrl + Alt + F12 to open Graphics control panel -> Options -> Information Center -> Save). Just attach those here and we can take a look.
Do you have access to another display? You can connect both the TV and the other display to your system, and then check the Intel® Graphics Control Panel and see if it is recognizing both monitors as being connected there.
If the system is not recognizing it as connected, you may have a faulty HDMI port on the motherboard.
After doing some more testing it turns out that when the DVI port is connected and it goes into the dual display mode the refresh rate for the HDMI/TV display was correctly being set to 24p. When the DVI port is disconnected and it is in the single display mode the refresh rate was being set to 60p for the TV which it couldn't sync to. After manually setting the refresh rate to 24p in single display mode it is working.
Thank you for your response.