[CAUTION: I am not a teacher:take the advice and or corrections offered in this post at your own risk.
If you doubt the information, please get a qualified opinion from one of the teachers on these forums.]
I would say more that it means the person entirely ignores an issue, not actual words, just as though they were deaf.
I would also say that it is used formally and for the imaginary representation of communication, not the actual words or sound (like a plea, or suggestion or request). I would go even further to say this saying implies the ears of many, and not the two ears of a person, but I can't prove that, it's just a feeling.
My suggestions to my girlfriend to change her ways continue to fall on deaf ears. (I wouldn't say this myself, as the g/f is one person)
The citizens were constantly opressed, but their plea to the UN for aid fell on deaf ears.
My wife was gravely ill, but my request to my supervisors for leave fell on deaf ears.
Your suggestion to the clergy that they should live by tested morals and not by proscribed rules will most certainly fall on deaf ears.