There is no way it can disallow root user any changes. This is linux.
Anyway, it is a very good idea not to upload/download web content as root. So you need a user for that. Create one. I guess you know how to create new users. Say, the user will be called webadmin.
So do this:
chown -R webadmin: /var/www
chmod -R 755 /var/www
and to be on the safe side:
chmod 755 /var
chmod 755 /
This will allow user webadmin modify, delete and add files in /var/www
Hope that helps. If not, could you please post exact commands you type and exact output you get.