Java is usually backwards compatible between versions so anything compiled on an old version should run fine on a newer JVM. In fact a lot of common libraries are compiled in as old a version as possible (usually Java 5 now a days) unless they need a newer feature to allow more people who are still stuck on old JVMs.
Having said that, there are a few gotchas you need to worry about. One problem I had on some Java 6 to 7 conversion was TreeMap with an initial value of null http://hariharanselvarajan-java.blogspot.com/2013/02/treemap-in-java-6-and-java-7.html
EDIT Here is a link to Oracle discussing what isn't compatible between 6 and 7 although I would imagine this only affects things that are recompiled: http://www.oracle.com/technetwork/java/javase/compatibility-417013.html