I guess I have my (docked) iPhone 4S set to never sleep because I have used this feature with Airfoil and never ran into a problem. For a while, you could also get an app called AirMusic that did the same thing. Unfortunately, it has been pulled, as well. The interesting thing is that this already works with these apps, so it would take no effort at all for Apple to add it to the OS.
In the meantime, you would need an Airplay receiver (Airport Express, some Airplay speaker with a headphone out, receiver with Airplay, etc..) connected to headphones or a DAC.
My dad has a Roku with the headphone in the remote and he says it eats through batteries at a blistering pace. I am not sure Apple would add this feature to the standard remote unless they can figure out a docking solution to recharge the remote when it isn't being used.