It might be a coincidence, but check your input device.
I had this same problem while trying to efile Minnesota state taxes. I looked through the forums and found no answers. Ultimately I called the support phone number. George spent an hour on the phone with me, trying multiple avenues, but without success. Finally, he sent me a new download of TurboTax Deluxe 2016.
I had already saved my files and so was able to uninstall my old version and open the existing files with the new download. When I got to the payment step for efiling state taxes, the Continue button still was non-responsive. For some reason--frustration, a hunch?--I unplugged the Wacom Bamboo tablet that I used as an input device (instead of a mouse or the laptop touchpad) throughout the TurboTax process,and clicked the "Continue" button using the laptop's built in touchpad and buttons. SUCCESS! I was able to complete efiling my state taxes within minutes.
I don't know if the solution was the change in input device or the fresh download of the program, but if you use an external input device -- mouse, pen tablet, etc.-- on a touchpad laptop, try clicking directly with the touchpad input.
