Environment: CentOS, Apache2, PHP 5.2
This is a horrible question, I can only apologise.

We're moving our site from our old host to a new one. I think the problems we've experienced are in the config.php file, specifically relating to the values given to variables like root_path, upload_path, etc (basically the ones that reference the file structure of the server).
I don't know PHP at all, but back in my VB days you could get around specifying actual paths by using a function like Server.MapPath and it'd figure it out for you. Anyway, I digress!
On our old host (for example) the root_path is:
I'm pretty sure that the file structure on the new server is different, however what I've been told by my host is that the equivelent path for root_path would be:/var/www/vhosts/ourdomain.com/httpdocs
I've highlighted in red part of what I think looks dubious, although I think the entire path looks too small. We have a VPS running DirectAdmin, which allows us to have multiple users and resellers. I'd definitely expect a longer path. SSH'ing as our admin user, the path from our lowest level directory is:/home/websiteusername/public_html
But all of these values result in the same errors when trying to reach the /admin screen:/var/www/vhosts/ourdomain.com/httpdocs
Apache error log says:Warning: opendir(/home/websiteusername/domains/ourdomain.com/public_html/admin/lang) [function.opendir]: failed to open dir: No such file or directory in /home/websiteusername/domains/ourdomain.com/public_html/admin/lang.php on line 33
Warning: readdir(): supplied argument is not a valid Directory resource in /home/websiteusername/domains/ourdomain.com/public_html/admin/lang.php on line 34
Warning: closedir(): supplied argument is not a valid Directory resource in /home/websiteusername/domains/ourdomain.com/public_html/admin/lang.php on line 39
Warning: Cannot modify header information - headers already sent by (output started at /home/websiteusername/domains/ourdomain.com/public_html/admin/lang.php:33) in /home/websiteusername/domains/ourdomain.com/public_html/admin/login.php on line 211
Warning: Cannot modify header information - headers already sent by (output started at /home/websiteusername/domains/ourdomain.com/public_html/admin/lang.php:33) in /home/websiteusername/domains/ourdomain.com/public_html/admin/login.php on line 212
Warning: include(/home/websiteusername/domains/ourdomain.com/public_html/admin/themes/default/login.php) [function.include]: failed to open stream: No such file or directory in /home/websiteusername/domains/ourdomain.com/public_html/admin/login.php on line 221
Warning: include() [function.include]: Failed opening '/home/websiteusername/domains/ourdomain.com/public_html/admin/themes/default/login.php' for inclusion (include_path='.:/usr/local/lib/php') in /home/websiteusername/domains/ourdomain.com/public_html/admin/login.php on line 221
(FYI, it's just running off an IP address whilst we get it sorted -- will repoint the DNS later. The root_url variable is set correctly for this.)[Fri Jan 09 16:09:17 2009] [warn] RewriteCond: NoCase option for non-regex pattern '-f' is not supported and will be ignored.
[Fri Jan 09 16:09:17 2009] [warn] RewriteCond: NoCase option for non-regex pattern '-d' is not supported and will be ignored.
[Fri Jan 09 16:09:17 2009] [error] [client 213.208.100.91] File does not exist: /var/www/html/home, referer: http://stagingipaddress/~websiteusernam ... /login.php
[Fri Jan 09 16:09:17 2009] [error] [client 213.208.100.91] File does not exist: /var/www/html/404.shtml, referer: http://stagingipaddress/~websiteusernam ... /login.php
[Fri Jan 09 16:09:23 2009] [warn] RewriteCond: NoCase option for non-regex pattern '-f' is not supported and will be ignored.
[Fri Jan 09 16:09:23 2009] [warn] RewriteCond: NoCase option for non-regex pattern '-d' is not supported and will be ignored.
[Fri Jan 09 16:09:24 2009] [warn] RewriteCond: NoCase option for non-regex pattern '-f' is not supported and will be ignored.
[Fri Jan 09 16:09:24 2009] [warn] RewriteCond: NoCase option for non-regex pattern '-d' is not supported and will be ignored.
Happy new year everyone!