I'm joining the party late, but I think I am qualified to answer this question based on personal experience. I was NOT born deaf and still hear pretty well in both the ears. In 2009, at the age of 26, I came down with "tinnitus" due to medical negligence. A retired army doctor syringed my ears the wrong way -- using a manually-operated "syringe" and using much greater force than was necessary. As a result, I began hearing incessant ringing in my left ear. Tinnitus, as I said. I was told to simply let the ringing be and "pay no attention it" -- well meaning advice that I failed to understand at that time.
Pretty soon, the ringing "spread" to my good, right ear as well. This, I'd learn later, was because my brain was trying to compensate the ringing noise in the left ear by creating a similar sound in the right ear. How on earth could the brain create a sound? Well, tinnitus has more to do with the brain than the ears, as we usually think of them.
How do we hear? Putting things simply, sound waves travel through the ear canal and make the ear drum vibrate. These vibrations are then passed on to thousands of tiny nerve hairs -- cochlea -- each tuned to a particular frequency. Turns out, tinnitus results from some of these tiny hairs misfiring in the absence of any external sound. Prolonged exposure to loud noises is the most common reason for cochlear damage.
Did I get better? Did the sound disappear? Yes and No. The sound didn't disappear, but over time, my brain simply got used to the ringing and phased it out -- a process that doctors call "habituation". In short, the ringing is still present 24/7, but I can't hear it because my brain doesn't attach any importance to it. Of course, when I try to deliberately tune in to my inner self, I can always hear it humming back to me.
So, what does my story have to do with the original question? Whether deaf people can hear "an inner voice" or not depends on the reason for their deafness. If their cochlea and hearing nerve are in working order, I'm sure they hear an inner voice. People born deaf may not even have a perception of what sound is, so they may not register that inner voice. People who were not born deaf, but went deaf later due to some reason, may notice that sound; but their brains (hopefully) tune it out over time. I've communicated extensively with those who have tinnitus and many of them are profoundly deaf.
So, the humming of tinnitus may be the only "sound" that some deaf people hear. That is what you probably call an "inner voice".
