No, a fundamental part of these jobs is communication and building trust.
You cannot do that with your face obscured. Even if it makes no difference to the sound volume and quality of speech, the simply fact that you cannot see another's persons face has huge repercussions for human communication, as there is great psychological value attached to faces and a huge amount of body language that is shown through facial expressions.
Even simpler than that, a hearing person's ability to recognise and understand words is not based solely on what they hear, and in important and long conversations draws a lot from being able to see the mouth movements of the person with whom you are speaking - 'lip reading' is unconsciously done by all us us, not just those with impaired hearing.
For those who say that a patient could ask someone to take it off:
No, the patient really couldn't. Not only is the patient role a frightening one wherein making these requests is very daunting (regardless of how nice the doctor is), in today's social environment, people could and absolutely would be accused of racism or some form of discrimination for doing so.