A relay that behaves like a simple one but doesn't continuously use power?
You're looking for an "Analog Switch".  They are basically solid-state relays that can pass analog signals.

There are many types, made by many manufacturers.  Here is a link to Maxim's MAX14689 to get you going.  It is a solid-state DPDT multiplexor suited to audio frequencies. It can be switched (and powered!) by a low-voltage signal from a microcontroller, or from your phone connector (if wired judiciously).