OS X shows "Sign in to use this application on this computer" dialogs when opening App Store applications
This same problem plagued me for months.  Any non-free mac store app would ask for a user name and password to access with the dialog box shown above.  When I lacked internet access, the app simply wouldn't load at all.  

After trying all sorts of solutions - deleting the apps and reinstalling, re-entering passwords and credit  card numbers in the app store itself, etc., I finally found something that solved the problem:

I reinstalled Mountain Lion from the rescue partition installed on my hard drive.  While this takes some time (about three hours), it isn't as drastic as it sounds.  All my apps and files and passwords were intact after the reinstallation, and for the past two weeks I haven't had to re-enter any mac app store passwords for the apps I've purchased.

here's a link to a Macworld article that explains the process of reinstalling the OS:

http://www.macworld.com/article/1167870/hands_on_with_mountain_lions_os_x_recovery_and_internet_recovery.html