Apple - Does the AirPods case need to be around while listening via an iOS Device?
No you don’t need the case to continue listening!
The case is only used for charging/storage and the initial (one time) pairing with your devices.
You can leave the case at home and get full usage out of the AirPods until the battery dies. This includes switching devices, monitoring battery levels, using Find My AirPods, etc.
I typically take the case with me unless I’m going for a quick jog/walk. I don’t recommend storing the AirPods anywhere other than the case or in your ears, to avoid losing them. Obviously the charging case prolongs the overall battery life as well.
Once you have paired the AirPods to an iOS device, you don't the case anymore to reconnect it to the device. The box helps in switching automatically to the nearby device and charging the AirPods.