The UI sounds on my Galaxy Nexus no longer play - how can I re-enable them?
This appears to be something of a bug, either in ICS or the Galaxy Nexus (though I am unsure which). Unbeknownst to me, there is a "system volume" level that controls, among other things, the output of UI sound effects. This volume level does not appear to be modifiable within the normal system settings, but you can modify it with third party apps.

Based on this information (which I obtained from an XDA thread), I downloaded Volume Control + from the Play Store and launched it. The very last slider on the interface was for "system volume", and it was set to a value of 0. Once I cranked it back up to the maximum, my UI sounds began playing normally again. I do not know what causes this to become muted inadvertently.