$ yum clean packages
That will delete any updates that have been downloaded and cached. It might free up some space, it might not. It depends if there's any updates downloaded and cached or not. Running this won't do any harm, the updates can always been downloaded again, and in fact they might already have been installed but the packages are still in the cache.
Other than that, well you got to delete something haven't you. First thing to do is figure out exactly what's taking up all your space. Ignore the contents of /opt because that's on a desperate partition and the contents of /mnt because that's mounted from somewhere else.
) is an excellent little tool for finding out what's using up your disk space, if you can find some space to download and compile it in
Other than that keep running 'du -sh *' in /var because that's the directory with the most in, then see what's the biggest directory in there, change in to that and run 'du -sh *' again and just keep looking around with that until you find something you can delete. But... don't delete anything unless you are 100% certain it's OK to do so. If in doubt, wait for whoever set it up to become available and have them fix it.