I could see this being really useful. There are many times I have needed questions answered by the doctor (e.g. I've been diagnosed with x - can I still do y safely) and the only way is with an appointment which wastes both our time. I wish my local GP would set aside 1 hour per day to answers questions via email or phone. It would make things much more efficient.
The issue with that, is there's no good way for the doctor to get paid for doing it. Prior to 2009 there was no billing code for phone calls or email and currently the rules around what is a "billable" consultation are complex and the coverage by insurers is far from uniform.
I was thinking more about my own healthcare system where the government pays the doctor. So his salary would just stay the same. In the US or other systems like it I can see how that would be an issue.