I had the exact problem today. My phone dial pad suddenly started making loud obnoxious tone when dialing a number. After looking to see if I could adjust tone and couldn't find a setting for that, I simply rebooted my phone which solved the problem. I've learned to always reboot my phone when I cannot find a quick solution to my problem.
Guess I am a year late in replying but it might help a poor soul later
Anyways I had the same problem today with my 5S. I cant find a permanent fix, but what worked for me is:
Open any game(which has sound effects)->Press volume down key. This will show 'Volume' being decreased. Whatever volume you set it to, now will be the volume for your keypad tones as well as games.
Hope this helps.