1. Check your build and see if there are any programs that are affected by upgrading Java. e.g there are certain software that still require 1.6. So before you push any upgrade to workstations check it on 1 workstation and see if anything is effected?
2. I know MAXIMO looses its functionality the moment you upgrade from Java 6 to Java 7.
3. All i can suggest, do rigorous testing before Java 7 is pushed.
Java 7 (especially 7u3) is still kinda young; the question is: what does it do for you that would justify the upgrade? Each upgrade costs time (during the actual upgrade and in testing your applications), so you need to balance that against the supposed gains.
e.g there are certain software that still require 1.6
That would be a very rare exception. Java is designed to be backwards compatible; if an application does not run on a newer Java version that should be considered a bug, and reported to the manufacturer.
Even if you aren't using any new language features, upgrading to the new version might be beneficial just because of improvements in the JVM, fixed bugs in libraries etc. Obviously as said before, you must make sure to test your app carefully