How can I use a Drupal 6 database in Drupal 7 without upgrading that database?
Simply put, you cannot unless you are willing to rewrite a lot of Drupal 7 parts.

One thing you can do, is to implement cron jobs to export changes in version-independent format like XML on source site, and import them from XML at target site.

Alternatively, you can use Services module and make one page call the other to perform certain tasks, like user login or article reading.