The profile manager will only open if there are no other Firefox windows open, so make sure Firefox is properly closed. You can add -no-remote to the shortcut to get around that.

Having said that, the built-in profile manager is probably going to be depreciated. Instead, a separate application will handle it. A pre-release version is available.

https://wiki.mozilla.org/Auto-tools/Projects/ProfileManager

As optional has described, if an instance of Firefox is already open, you have to add -no-remote: firefox.exe -p -no-remote

Firefox Commands

You can use -no-remote at anytime to open the profile manager.
You only can't open the profile manager if the -no-remote switch isn't specified and Firefox is running.

See also:

• http://kb.mozillazine.org/Using_multiple_profiles_-_Firefox
• http://kb.mozillazine.org/Opening_a_new_instance_of_Firefox_with_another_profile