Hi,
Currently have a script to rsync all the files from local to remote

rsync -zvr --exclude-from='/usr/local/sysadm/exclude-list.txt' /usr/local/apache24/ test@192.168.1.1:/tmp/Backup/$DATE

But the disk space usage is high in the remote server. To reduce space, I need to zip or tar all the files....

I need to rsync the files first and then remotes zip or tar all the folders....

I am using the below command to zip the files remotely

ssh test@192.168.1.1 gzip /tmp/Backup/$DATE/*

but I do not know how to zip the folders remotely. Also the zip or tar should remove the existing folders and only the zip or tar must be present in the remote server

none of these tools will remove files during operation. That means after a successful tar/gzip/whatever can be the source files deleted.
if you want to rsync gzipped files you need to use a version of gzip with the option --rsyncable.

Why you execute the zip from remote. Instead, add a cronjob (find and zip all the files)

I do not need all the folders to be zipped. I need only selected folders to be zipped/tar.

If I try to tar locally by excluding some folders, I would be left behind with the tar file, for this I need to put a separate cron to remove them, which I don't want. So I thought of zipping them remotely in the same script....

What about rsnapshot which will avoid duplication between the backups taken on different days?

If you use zip rather than gzip, you could have it remove everything, e.g.

The zip archive /tmp/Backup/${DATE}.zip would contain everything that was in /tmp/Backup/$DATE previously. After the zip command is finished /tmp/Backup/$DATE and its subdirectories and files would be removed.

Alternatively if you just want to use gzip and do not care about everything being in a single archive, you could:

For better space saving use xz instead:

EDIT: Actually, given this is a backup I would use bzip2 rather than gzip or xz. Although the result will not be as small as xz compression, bzip2 has better recovery tools (in the form of bzip2recover), should any of the files get damaged or corrupted in the future.

P.S. I'd also recommend lbzip2 instead of bzip2 for a quicker result.