Great question with some good answers.
Everyone should know running wired headphones should save considerable power versus the built-in speaker.
But wired headphones vs Bluetooth... a quick google...
Wired headphones should consume less than 1mW to give ample volume. ( example sensitivity of 100dB SPL / mW)
Bluetooth transmits with variable power -- from 0.01mW to 100mW -- depends on range, I would imagine, the power adjusts to keep data flowing without errors. Since we use the headphones pretty close to the transmitter, I bet it's in the middle magnitude...somewhere like ... 1mW.
Both are, relatively speaking, drawing peanuts vs. the rest of the machine. Bluetooth wireless *could* suck down a lot more power if there's signal blockage or interference, so the *safest* answer is to run wired. In reality -- run what you want, probably won't matter most of the time. Bluetooth draw may be location dependent. Any anecdotal experiences would be interesting to hear.