This depends on your headphone driver (and it's ability to source current), but I'm thinking it may power the LED. A few things you should know however:
Generally, audio signals are not DC. You will have to create a very specific audio file to play in order to accomplish something like morse or binary code. Do this in WAV format, or something uncompressed, as MP3 and other audio compressions encode frequency content, and will ignore the bias that you are looking to create.
A normal LED drop is not 0.7V, it depends on the type of LED you buy. You can expect LED drops to be approximately 1.7 to 3.3V, with a current draw of approximately 20mA. Try to find an LED with a lower voltage drop, this will allow it to turn on in a lower range. Again, your headphone driver may not be able to accomplish this.
If you have a multimeter handy, try creating a WAV file with the maximum bias on it, and measure the DC voltage on your headphone driver with a small known resistive load on it (try 100 ohms, as most mid-range headphones are around this impedance). This can let you know the current/voltage your driver is capable of, so you can make the best resistor choice.