I have had the same problem with Zimbra Desktop since at least Fedora 22. I just managed to work around it by doing the following. Note: I do not claim that my solution is correct or that anyone else should follow it.
- Inside the newly-downloaded/extracted directory containing the Zimbra app, open the script install.pl in a text editor and remove the "exit;" from the line: "else { print "Java SE Runtime Environment (JRE) not found. Please install JRE 1.6 or later.\n"; exit;}". This will prevent the installation from exiting, if the installer does not detect the appropriate java version in your system.
- Install the app by running the script, as explained in the Zimbra documentation.
- Run the user-install.pl according to the documentation.
- If you have followed beduine's answer above, it means that you will have the 32-bit JDK version in your /opt/ directory. You will need to point to this java version in the following two files: "/opt/zimbra/zdesktop/data/bin/zdesktop" and "/home/user/zdesktop/bin/zdesktop". To do this, simply comment out the line JAVA=which java and add the line JAVA="/opt/jdk_directory/jre/bin/java".
You may need to install some extra packages, so it is probably better to run Zimbra from the terminal and look for missing packages-related warnings/errors.
Hope this helps.