To answer your question: yes, your Beats need to be powered on and within range of either your iPhone or another Apple device that’s part of the Find My network in order to update their location. They don’t have to be actively connected to your iPhone via Bluetooth at all times, but they do need enough battery power to be discoverable. If they’re turned off, the battery dies, or the thief disables them, their location will stop updating until they come back online.
The Find My app is your only official tracking method for Beats headphones. Unlike AirTags, Beats don’t have ultra-wideband tracking or Lost Mode features. They can piggyback on Apple’s Find My network, meaning if someone else with an Apple device comes within range of your headphones, their location could update. However, if they’re powered down or out of range, the app will just show their last known location.
If you haven’t already, I’d recommend marking them as Lost in the Find My app. This way, if they do come back online, you’ll get a notification. Beyond that, your options are limited to keeping an eye on the Find My map and hoping the headphones reappear when powered up.