As it turns out, it is not trivial to control the audio volume of an Android device using ADB. At the time of writing, the only way appears to be using the service tool. Actually, the service command allows to "connect" to a number of services (104 on Android 6.0.1) and invoke functions. Not knowing much about this tool, I managed to completely mute all sounds and speakers of my Nexus 5, and I was stuck without any sound for quite some time. I did not find a way to unmute the sound from within the system UI, so I got to dive a little deeper into this.
If you know which service you want to use, you then need to find its interface declaration. The command
service list
gives you a list of all services with the associated interfaces, if applicable:
...
26 backup: [android.app.backup.IBackupManager]