When Do Not Disturb is set on iPhone, what will fire off and what will not?
Source.
Mastering DND

Do Not Disturb effectively silences almost all notifications while you aren’t actively using your phone. You can set it so your phone won’t ring or buzz when you receive phone calls, new text messages, new email, or new invitations from annoying “friends” who want to challenge you to a game of Candy Crush or whatever.
Certain notifications get through no matter what, such as timers and alarms set through the Clock app. You may still receive various government alerts too, such as emergency alerts and AMBER alerts.

Continue to the source article for more detailed info on customizing DND to get the most from the feature.