We merrily got the new JDK and slowly promoted it out from dev to qa to staging and then to two servers in prod.
Naturally, there was a problem that wasn't discovered until we got to prod, but on testing other environments was present everywhere. What was the problem? there was a security exception preventing the server from accessing an https URL now over SSL.
Huh? Well, Java is very very strict about SSL / HTTPS connections, so if you try to use a Java program to connect to a server over SSL using an HTTPS URL, and that server doesn't have an SSL certificate that is signed by a CA in your JDK cacerts file, you'll have problems. The normal solution for this is to use keytool to add the CA to the JDK's cacerts file, or to make a keystore using keytool with the certificate in it.
Additionally, it is possible that the java.policy file itself has been changed to grant (or revoke) certain permissions from the JDK.
Both of these files live inside the JDK, so if you upgrade the JDK but forget to copy those files over (like we did) you're going to have some problems.
Finally, for very very old JDKs (1.3.1 and older, pity me) note that you will have other issues if you don't download the JCE and JSSE since without those, the old JDK can't do strong crypto and you'll have trouble too.
In order to discover whether you need to care about this or not, you should like at JAVA_HOME/jre/lib/ext (the JSSE and JCE files go in there) and JAVA_HOME/jre/lib/security/ to see if any files have been updated. Or you can use something like a "find . -mtime