No, git will not track database changes. Features however can handle much of the configuration (that is typically contained in the database) in code, which can then be managed with git.

If you want to create content on your dev or staging site and push it to production, check out the deploy module: http://drupal.org/project/deploy

Tracking changes to nodes in production is probably best left to Drupal's built in node versioning system. Diff (http://drupal.org/project/diff) is a handy module to track changes between revisions and if you want a little more advanced workflow check out revisioning (http://drupal.org/project/revisioning) and workflow (http://drupal.org/project/workflow). Good luck!