So, apparently I did make a bad assumption. You do not need Push on Dovecot for notifications to work on iPhones. However, the iPhone has a bunch of settings in various places which can prevent it from working. So, this is a software/settings issue on iPhones, not something that has changed causing a need to change Dovecot.
Thanks so much for all the replies and information. I was certainly about to waste a lot of time.