HDMI port damaged

The HDMI port may be damaged

Please also check you haven't accidentally plugged it into the HDMI in port

Sounds like you may have a damaged HDMI cable, try changing that, if it is a Monster HDMI cable, they have a lifetime warranty and will replace the cable for you. The HDMI port on the television may be damaged, which can happen, I have seen it happen myself,
try a different HDMI port and do not use a HDMI connector box that allows many HDMI connections to be put into one HDMI port.

I would also add to check the HDMI cable for any damage and both the HDMI port out to the TV and the HDMI port on your TV to see if there is any damage. Also if you have any other free HDMI ports on the TV plug it into them and see if the problem persists.

Also try to see if there is a build up of material (dust) on the cables ends and HDMI ports. There are several ways to clean HDMI cables and ports. Just look up how to clean them properly.

If neither of the above works, check the HDMI port visually on the console for damage. You say you've tried several ports on the tv with the same result, so this points to either the console or the cable being damaged. If the HDMI port looks ok, possible
that the cable is damaged, check the plugs at each end for misaligned pins, or try a different cable.

Sorry to hear it. If your console has composite cables then you can connect them. Otherwise, if you have a 360 S then you can send it in to get repaired.