Drupal 7 update “dom” requirements error solution: If you go to update a Drupal 7 website and get a “dom” requirements error, it turns out that you need to install the php-xml package, at least on a Linux system. The solution for me involved running these two commands:
yum install php-xml
service httpd restart
FWIW, this was not a requirement with early Drupal 7 versions, but was added somewhere down the road.