So apparently the Equalizer (or audio) UI component on the lockscreen causes my device to completely become unresponsive to touch from about the middle of the screen down. This started when I got the Lollipop (CyanogenMod 12) OTA update, and I can correlate the behavior to that (restart the device, don't play audio - no issue).
Can I disable that UI component for the lockscreen (or the Equalizer in CM12 if it comes to that) until this is fixed? My device is a OnePlus A0001.
Settings > Lockscreen > Turn off "Display music visualizer"
The option to disable it only exists on some of the newer nightly builds. You won't see it on CyanogenOS 12.