Java team takes extreme level of care while adding new features/APIs so that old code should not break. Theoretically, code which worked with JDK 6 should work with JDK 7 without a single issue.
I've migrated a Core Java application from JDK 5 (i.e. 1.5) to JDK 6 and didn't have a single issue. All you'll have to do is - install JDK 7, recompile/rebuild your application (which will be flawless) and make some test runs (here too, I don't see any possibility of issues).
What kind of application do you have? Is it Core Java, or dependent on anything else like web/application server?