How do I modify Drupal so that ONLY relative URLs are used through the whole site? I have a client who says that he is unable to view his company's website from inside his company's building. Their network engineers have asked to make sure that DOMAIN.COM is not included in any of the URLs generated by Drupal. 

They want only relative paths everywhere in the code. Any insights?
You can alter all Drupal generated URLs to make them root relative using hook_file_url_alter

function mymodule_file_url_alter(&amp;$uri) {
  global $base_path;

  $scheme = file_uri_scheme($uri);

  if (!$scheme || $scheme == 'public') {
    if (!$scheme) {
      $path = $uri;
    }
    else {
      $wrapper = file_stream_wrapper_get_instance_by_scheme($scheme);
      $path = $wrapper-&gt;getDirectoryPath() . '/' . file_uri_target($uri);
    }

    // Clean up Windows paths.
    $path = str_replace('\\', '/', $path);

    $uri = $base_path . ltrim($path, '/');
  }
}


For links in content you can apply the filter supplied by Pathologic