On battery operated radios: If you turn the volume up, does that pull more juice from the batteries?

yes

Absolutely. You can’t increase the volume without using more energy.

Small radios don’‘t take a lot of power though. I have a shower radio with three C cells. I use it 30 to 60 minutes a day, and the batteries last 6 months or more.

Yes, and here is why-
A conventional speaker creates sound by vibrating a membrane. The membrane is pulled by a magnet on the back. The magnet is surrounded by a wire coil. When the current through the coil changes, the magnetism generated by the coil pulls the magnet and the membrane. The larger the change, the louder the sound. To generate large variations, you need more current. This current comes from the battery.

@PhiNotPi – Saved me a lot of typing. and used less profanity