I am experiencing an IPv6 problems with my iOS apps. Initially I submitted a few days before June 1st not knowing about IPv6 requirement (https://developer.apple.com/news/?id=0
For what it's worth, I saw this as well and as discussed here, there's clearly a discrepancy between the documentation as of 6/14/16 and the App Review Team's setup.
As I suspected this did not have to do with the advertising and analytics code, as commenters suggested. It had to do with server compatibility with IPv6. I use Linode. I had my linux server admin upgrade to IPv6 for me on my server. If advertising or analytics failed to connect, they would just silently fail, there is no issue with game play. The problem was talking to my game server, which didn't work under IPv6 only.
According to answer: https://stackoverflow.com/a/37689996/211457 I used this site to test my site's compatibility: http://ready.chair6.net/ - http://ready.chair6.net/?url=https://greenrobot.com
Interestingly, Apple.com, Google.com, Facebook.com, Twitter.com are all non compliant. The only other domain I could find that was totally compliant like my site now is is arin.net.
After making these changes my app is now approved. Note I also added some error checking regarding not having a connection and displaying an appropriate error screen. But since that other Stackoverflow poster also had IPv6 server problems, and it doesn't make a lot of sense that just adding error checking would cause the app to suddenly connect, I am pretty convinced it was a server problem.
Another interesting thing is I talked to an iOS developer on the phone who had an update approved after June 1st. They are using AWS and had no idea what I was talking about regarding AWS not working under IPv6. Very strange, unless Apple is only requiring IPv6 only on a percentage of apps.
edit: New information: from a comment below: "If your server has an IPv6 record (i.e. claims to support IPv6), but your server's IPv6 support is broken, then it won't work on IPv6 obviously. But if your sever doesn't have an IPv6 record at all (i.e. claims to IPv4-only), it should work without any problems."