"Do Not Disturb" mode only mutes notifications while the iPhone's screen is locked. The text on the DnD settings page in iOS specifically says "When Do Not Disturb is enabled calls and alerts that arrive WHILE LOCKED will be silenced, and a moon icon will appear in the status bar" (emphasis mine)[1].
I've been bitten by this during a demo: I still received texts (from a co-worker) even though I had DnD enabled.