Disclaimer: Boring Video.
This happened to me tonight. I was trying to setup instagram with my facebook account, but when I got to the login as my facebook id, I tried clicking the button to select my facebook id, but NOTHING happened.
Again, I was stuck at the facebook login for instagram. All I needed to do was click "Continue as..." but nothing worked. It was like none of my finger taps were registering.
The fix for me was rebooting the iphone.
I tried force closing the app as well as updating it, but it did nothing.
Remember.. when you have app problems,.. the easiest thing to do is to reboot your phone! voila. problem fixed.
