Yes, but this may or may not be a significant amount.
In any device there will be some “background” level of power draw which is independent of the volume (such as the CD motor, or cassette drive, or MP3-playing microprocessor) plus a variable amount of power that depends on the volume.
With a small analogue FM radio receiver it’s possible that the receiver-part draws very little power, and most of the power goes to the speaker, so the volume-setting will make a significant difference (easily halving or doubling the battery life depending on the volume). Also in one example the three AA batteries lasted for 100’s hours on headphones, but only 20-30 hours on speaker. It takes a lot less power to drive headphones than a speaker.
If it’s a modern digital radio receiver, MP3 player, or mobile-phone (with WiFi etc), driving headphones, then it’s likely that the “background” power-consumption is much higher than the power to the headphones, so the volume will make little difference.
Latest Answers