Turtle beach's hiss troubleshooting. This isn't what I experienced, but it might help some people that reach this question. To sum up, three possibilities:
This is an amplified headset. Amplifiers amplify all sound (including noise) originating from the source. Solve by turning down the amplifier and turning up the source. (this isn't you, as you hear the hum with a silent source).
XBox controller talkback wire. Disconnect it to see if the hiss goes away. (no long term solution presented other than leaving the wire disconnected).
Microphone noise being sent back to through the headset to your ears (I think if this was happening, it would be obvious). Mute the mic to diagnose.
So I did some more googling and found out about Ground Loops.
Ground loops created by improperly designed or improperly installed equipment are a major cause of noise and interference in audio and video systems. They can also create an electric shock hazard, since ostensibly "grounded" parts of the equipment, which are often accessible to users, are not at ground potential.
And then I found Turtle Beach's Ground Loop article.
Perhaps an electrical engineer can give a better description of the problem, but from what I understand, the headset is connected to two devices (the console and the tv). These devices don't agree on what zero signal is, so when the TV's zero signal is viewed from the console's zero, it's not zero any more... not zero = signal = the noise you hear.
The articles recommend introducing a two prong adapter to diagnose the issue, but no long term solution is given.
You may also want to watch this guy talking about ground loops: