Clicking on the "View unanswered posts" link on the board index page generates a database query that can't use any indexes, requiring a full table scan of phpbb_topics.
The code is in search.php line 385. The query looks like this:
mysql> EXPLAIN SELECT DISTINCT t.topic_last_post_time, p.topic_id FROM phpbb_posts p, phpbb_topics t WHERE t.topic_replies = 0 AND t.topic_moved_id = 0 AND p.topic_id = t.topic_id AND p.post_approved = 1 AND p.forum_id NOT IN (17, 22, 28, 29) ORDER BY t.topic_last_post_time DESC LIMIT 1001 \G
Extra: Using where; Using temporary; Using filesort
Extra: Using where
2 rows in set (0.00 sec)
On my forum, this isn't called very often, but can take 30 seconds or more when it is.
FYI, the "Tapatalk" mobile plugin copies this code, so also generates these slow queries.