As of Java 7 update 51 (7u51), there is a requirement for a new Permissions attribute (introduced in 7u25) that prevents the FileMaker Server 12 Admin Console from running. This also affects the version 11, 10, and 9 Admin Console and probably other FileMaker products utilizing Java.
After a warning that the application will be blocked by future versions of Java you get a generic "Application Error" declaring that Java was "Unable to launch the application.". Diving into the details will turn up the precise reason as:
Missing required Permissions manifest attribute in main jar: http://fm-server:16000/admin-client-lib/admin-console-client.jar.
FMS 12 is updated to the latest version at this time and the Permissions attribute has not been added to the Admin Console jar by FileMaker, Inc.