How much will using headphones increase the battery life?
Headphones take much less current to drive than the speaker in the detector. This fact by itself would tend to show that you will increase your battery life by using headphones. But you have to remember that even though the detector is not making any noises, the electronic circuits are still running. A detector that generates a square wave or has a display will be using more power than a detector that is using a sinusoidal wave and has no display. The increase in battery life will depend on your detector and hunting style.