Just wanted to confirm that our house's Roku 3 is exhibiting the exact same behavior with Disney Now app, and also today with the Amazon Prime Video app. For both, you launch the app from the home screen, it boots into the app to some degree, and then instantly dumps back to the home screen. For Disney Now, it typically gets to the menu screen where you can try to choose the Disney show to watch, before dumping out. With Amazon Prime, it gets through one or two of the splash/loading screens, then dumps out.
I uninstalled the Disney Now app, did a System Restart from the Settings menu (not an actual power off), then reinstalled and it worked for 1-2 shows. We exited the Disney Now app, came back later, and it's back.
I haven't reinstalled Amazon Prime Video yet, so I may uninstall Disney Now again, do a full unplug power off cycle, then reinstall and see if it changes anything. With twin 2 year olds, not a great time for this to start happening...