Question: Which one consumes more battery, listening to music through the speaker of a phone or through the earphones?

Answer: Commonly speaking, of course the speaker consumes more. You can even feel the vibration if you put your hand near it. The power needed to drive a bigger diaphragm have to be higher otherwise it just doesn’t work at all. Why “commonly speaking” because I'm not sure what the conclusion will be if you turn down the volume to merely mute.

It depends on the headphones, some of them especially the high quality audiophile units are fitted with DACs and usually have powerful speakers that draw a lot of power.