How to display smart group search results to unauthenticated users?
I will assist Xavier answer, as I don't have reputation to comment.

The user permission can be found navigating to Administer>User and Permissions>Permissions>WordPress Access Control
URL: www.yoursite.com/wp-admin/admin.php?page=CiviCRM&amp;q=civicrm/admin/access/wp-permissions&amp;reset=1

CiviCRM: profile listings and forms has to checked for anonymous user, if you have custom data in your profile, as Xavier says you will need to enable it as well.

Also make sure the profile has both "Standalone form or directory" and "Search view" checked. 

And last, make sure your Smart group has "Public pages" as visibility, in the group settings.

Groups URL:
www.yoursite.com/wp-admin/admin.php?page=CiviCRM&amp;q=civicrm/group&amp;reset=1

Hope that helps.