To get this to work I had to set the LDAP version to 3 using ldap_set_option. Here is an example that might help:
$TheDN = "cn=john smith,ou=users,dc=acme,dc=com";
$newRDN = "cn=bill brown";
$newParent = "ou=users,dc=acme,dc=com";
ldap_set_option($ds,LDAP_OPT_PROTOCOL_VERSION,3);
@$result = ldap_rename($ds, $TheDN, $newRDN, $newParent, TRUE);