Depends on how you've installed and configured it. If you've used one of the later Jira installers on some operating systems, then it's installed it's own Java environment and your system update is not going to affect it.
Have a look at the system information page, that should help you determine what you've installed.
I'd not try to update it. I'd install 1.7 on the server, and then change the setartup scripts to point at the new install. I've done it a couple of times by changing the setenv.sh file which sets the environment, you include the JAVA_HOME environment variable
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