You need to check the files which are consuming space in /var file system
# du -Sx /var |sort -nr |head -n 10
It will give you the list of top 10 large sized files. But don't zipped them directly, there you need to check whether any OS/ any application is writing log files or not. If it is writing the logs then log rotation need to configure.
For OS logs, use below link for reference
or for application, you need to check the application configuration.