I have a site at:

sub.site.com

I want to other domains to point to it. On the old server, everything was setup through cpanel. On the new server, I have the vhosts file as:

<VirtualHost sub.site.com:80>
ServerAdmin webmaster@dummy-host.example.com
DocumentRoot /home/sites/site
ServerName sub.site.com
ErrorLog logs/dummy-host.example.com-error_log
CustomLog logs/dummy-host.example.com-access_log common
</VirtualHost>
<VirtualHost domain.com:80>
ServerAdmin webmaster@dummy-host.example.com
DocumentRoot /home/sites/site
ServerName domain.com
ErrorLog logs/dummy-host.example.com-error_log
CustomLog logs/dummy-host.example.com-access_log common
</VirtualHost>
<VirtualHost domain.net:80>
ServerAdmin webmaster@dummy-host.example.com
DocumentRoot /home/sites/site
ServerName domain.net
ErrorLog logs/dummy-host.example.com-error_log
CustomLog logs/dummy-host.example.com-access_log common
</VirtualHost>

It seems like this should work, because all three are pulling from /home/sites/site

But it doesn't work. It also gives a message when I restart apache:

VirtualHost domain.com:80 overlaps with VirtualHost domain.net:80, the first has precedence, perhaps you need a NameVirtualHost directive

Hi,

As the message says, you need a NameVirtualHost directive. The usual syntax is
Using the above notation you need to change the opening VirtualHost tag of the 3 vhosts, like this:
Regards

Will it matter that there are other vhosts on the server? It seems like NameVirtualHost *:80 would apply to everything.

You use the generic NameVirtualHost *:80 and later on when it come to define the vhosts, you use:
and so on for any virtual host you want to add
Running
should give you details on vhosts.

Regards