In my case - and hard to say if it's applicable to yours, though I'm betting some form of it is - I'm running on a VPS with Plesk 8.1.1.

I was getting this exact same error during install (also received it for some other CMSs I was evaluating).

The issue turned out to be that Plesk uses a different httpd.include file for every virtual host. If you are a VPS user, you can augment this by creating (or editing) a vhost.conf file and adding the following for the path your vhost is on:

<Directory /var/www/vhosts/yoursitehere.com/httpdocs>
php_admin_value open_basedir none
</Directory>

or, alternatively, if you are not on a VPS, you can edit this in Apache's httpd.conf file directly by using

<Directory>
php_admin_value open_basedir none
</Directory>

You are supposed to be able to specify one or more explicit paths following the open_basedir attribute using standard ":" convention, but I could never get it to work by doing so. With a path in the mix, the installers (not just SS) always seemed to throw the fatal error.

Now, if I could get past the "Undefined Index" errors following my SilverStripe install(s) everything would be great. ;-)

Best of luck,
Jorgen