I have an existing drupal page and I like to update to latest 7.34 version. I read, that it's possible using drush pm-update command. Can this command cause some problems on the site? It's on production server. Thanks.
As any other software, it's possible that drush could cause some issues. But anyway, drush is the recommended way to manage Drupal and that applies to pm-update command as well. You should consider making your own sandbox (dev environment) in order to test your actions before deploying your changes to any production server.