If there's a problem with PowerPress we'll fix it, no need to imply otherwise. The problem though is the specific error "MySQL server has gone away" is a MySQL server error (not an error with the application). If you read this page (http://dev.mysql.com/doc/refman/5.0/en/gone-away.html), it gives you extensive details to figure out how to solve the problem. They are all specific to MySQL, not the application generating queries against MySQL.
The specific query you are having an issue with is in WordPress itself, in file wp-includes/taxonomy.php, starts on line 1971 through line 1979 in WordPress version 3.6. This is called by function wp_get_post_tags. PowerPress does call wp_get_post_tags in function powerpress_rss2_item. This is why you see it in the error message after the query error. The problem though is not in PowerPress. You are right, you can turn off PowerPress and your problem will go away. But you should not receive this error in the first place. Some day you may run into this same problem with something else because it's the query made by the function wp_get_object_terms() causing your problems, not a query made by PowerPress.
So first question, is it always the same exact query? Specifically, where it says ... AND tr.object_id IN (5) ....? The 5 is an ID, presumably to a blog post. I would edit that blog post and look for abnormalities like an excessive number of tags, etc...
Another thing to try, take the query above, go into phpMyAdmin on your server, and execute the query once to see how responsive the query is (how many results it returns, etc...) Then run the same query again, with the word "EXPLAIN " in front if it. It will give you some facts on what index it is using, etc.. Go ahead and copy and paste those EXPLAIN results (both rows, the header row and the actual result row) here, that may give me an idea what the problem is.
My only guess is that there's an excessively large number of tags for the post with ID 5, the query takes longer than X seconds to return results, and your MySQL server kills the query before it gets a chance to return results. Maybe because of a missing index in one of your tables. Again, this is a guess, not the actual answer to your problem.