Read this post if you have problems.

The QuickJava extension doesn't actually remember (or forget) any settings because it simply loads the settings from Firefox upon loading.  Because it's functionality is so simple most functionality bugs are usually not a problem with QuickJava but rather with a plugin (such as Java or Silverlight) or Firefox itself.

To verify what is causing the problem, please follow the following steps (please read and follow through the whole thing before asking for help).

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Note: All references to "Restart" mean restart firefox, ALL FIREFOX WINDOWS must be closed (including the addons and download manager). You should NOT need to restart the entire computer.

Please try this: Tools -> Addons and Disable all of the addons, go to the Plugins tab and make sure plugins related to the problem you are experiencing are enabled (for Java there are usually 2). Restart Firefox.

Please note: The Add-On -> Plugin window does not update dynamically, you must close it and re-open it to get the current enabled/disabled status of the plugins if you change it from QuickJava. This is important to remember moving forward.

After manually making sure everything is enabled, find a test page for the plugin you are experiencing problems for.  These pages can easily be found by googling "test java" or "test silverlight", ect.  For example you can test Java here:
http://www.java.com/en/download/help/testvm.xml

If this test fails then you have a broader issue that I can't really help you with but there should be information on that page that can help. You can not proceed with any more tests until you can get the relevant tests to run properly.  After those tests are running properly, re-test QuickJava as your problem will most likely be resolved.

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Restart, Enable QuickJava, restart. Click the J, verify the plugin disabled, restart make sure it's still disabled. Click the J again, verify the plugin enabled, restart and make sure it's still enabled. Try clicking the J one more time, restart and make sure it's disabled still.

If it works fine w/ ONLY QuickJava enabled then it may be another one of your add-ons. You can try enabling them one (or a few) at a time and re-testing each time to find the problem and contact the developer of the problematic add on for further help.

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If you are still having problems disable QuickJava, restart.

Tools -> Addons -> Plugins and make sure BOTH Java plugins are enabled. Disable both of them, restart firefox and check to make sure they are still both disabled. Set them both to enabled and restart FF and check them again.

If they are not holding their settings at this point then your browser may have a corrupt install or something, contact the mozilla support.

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If it was working with all extensions disabled, and not working when ONLY QuickJava is enabled then we'll need to talk more.

If this helps you find a solution, please post here so that others can read it.

A previous user's response:
"Thanks for the very complete diagnostic check list.
Actually, QuickJava runs well and enable/disable correctly the Java plugins. But only one of the two plugins stay disabled between two restarts. The "Java Deployment Toolkit" is always re-enabled at startup. So I will investigate that problem.
Thanks again for your support."
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Update for newer versions of FireFox:

To manually enable javascript that has been removed by the FireFox options page, you need to put "about:config" in the firefox url bar and then search for javascript.enabled and set it to true.
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 Update for Mac users from E Jonsson (via email):
An extra tip if you are having problems getting Java to work (regardless of QuickJava).
"I have subsequently found out that Firefox seems to be sensitive to whether you open in 32 or 64bit mode. The problem gets overcome by highlighting the Firefox application in Finder and pressing cmd-i for Information. Then untick the ‘open in 32 bit mode’ option. For some reason Java is not that sensitive in Safari, though. It works in both modes."