I noticed this recently when I disconnected my Microsoft account from a Windows 8.1 machine. With the help of VS2013 debugger, I found that it was caused by a file permission issue, the metro app cannot access its local settings file (which is located: C:\Users\<username>\AppData\Local\Packages\<appfolder>\Settings\Settings.dat), and also several other files under its app folder.
Solution is pretty straightforward:
1. Open explorer and navigate to C:\Users\<username>\AppData\Local\
2. Right click on "Packages" folder, select property, then security tab.
3. On security page, click Edit which brings us to "Permissions for Packages" dialog, click "Add..."
4. On "Select User, Computers, Service Account, or Groups" dialog, click Advanced...
5. Change "Locations" to your local PC (in case you are in a domain), then OK.
6. Now we back to "Select User, Computers, Service Account, or Groups" dialog, and click "Find Now". You should notice an item named "ALL APPLICATION PACKAGES", ensure it is selected, then click "OK", back to "Select Users or Groups" dialog.
7. Now keep clicking "OK" to dismiss all dialog.
8. Try to launch Metro App again, it should work now.