These are the potential issues you could face:
1) /etc/chkserv.d/httpd
This file contains monitoring information for Apache. Because Varnish must listen on port 80, we make Apache listen on port 82 instead. So the above file must be changed from this:
service[httpd]=80,GET / HTTP/1.0,HTTP/1..,/scripts/checkmaxclients;killall -TERM httpd;sleep 2;killall -9 httpd;/etc/rc.d/init.d/httpd stop;/etc/rc.d/init.d/httpd startssl;/usr/local/apache/bin/apachectl startssl;/etc/rc.d/init.d/httpd start;/usr/local/apache/bin/apachectl start
to this:
service[httpd]=82,GET / HTTP/1.0,HTTP/1..,/scripts/checkmaxclients;killall -TERM httpd;sleep 2;killall -9 httpd;/etc/rc.d/init.d/httpd stop;/etc/rc.d/init.d/httpd startssl;/usr/local/apache/bin/apachectl startssl;/etc/rc.d/init.d/httpd start;/usr/local/apache/bin/apachectl start
The only item changing being the port number 80 to 82.
2) Varnish is not starting
The number one reason for Varnish not being able to start, especially on 32-bit, is the lack of memory. Be sure to have plenty of memory and make sure the bin file size is set to the minimum. You can change the bin file size in WHM -> UNIXY.NET Varnish Plugin -> Advanced Configuration. Try setting it to "16M" and submit. Then restart Varnish from the management page.
There's also the issue with the lack of addressable space (VM). Should this be the case, be sure sure to:
1) rm -f /home/varnish/varnish.bin
2) from WHM -> cPanel Varnish -> Advanced Configuration -> and set the cache to 32M
3) Domains displaying the cPanel default page
The quickest way to resolve this issue is to reinstall the plugin. This usually caused after adding new IP addresses to cPanel.
4) Varnish is running out of memory
If Varnish runs out of memory, and assuming you have plenty of spare physical memory available, be sure to increase the allocatable memory pool that Varnish has access to.WHM -> UNIXY.NET Varnish Plugin -> Advanced Configuration -> Memory Cache -> 2G
5)SyntaxError: Non-ASCII character '\xf2' in file installvarnish on line 1, but no encoding declared; see http://www.python.org/peps/pep-0263.html
The above error pops up when running the installer (/usr/bin/python installvarnish). The error means you're installing the wrong package. If you've downloaded the CentOS5 package, you might want to try the CentOS6 one instead and vice versa.