I've a set of DC ANRs.
When the batteries go, there is more engine external noise which makes hearing ATC a little harder. The clarity of the audio isn't reduced, it's just that the aircraft noise is louder.
If the ANR is left turned on while the batteries are gone, I get an anoying clicking type sound (not dissimilar to the ticking often heard on a radio when a mobile phone is nearby) but again the clarity of the audio itself is not disminished.
dp