Q: Do Not Disturb malfunction?

Please note that the Do-not-Disturb feature is only functional when the iPhone is in sleep mode.

Try resetting all settings. that doesn't fix it, Back up and restore using itunes.

http://support.apple.com/kb/HT5463

Additional Information

Optional settings for Do Not Disturb:

• Scheduled: Automatically enable Do Not Disturb between the hours you specify.
• Allow Calls From: Allow calls from everyone, no one, your favorites, or specific contact groups stored on your device or your iCloud account.
• Repeated Calls: If someone calls you twice within three minutes, the call will not be silenced.

You're welcome.

To turn ON/OFF "Do Not Disturb" - on the left side of the phone (where the volume buttons are); the top button is the "Do Not Disturb" button.  If in "up" position (toward screen) the "Do Not Disturb" feature is OFF.  If in the "down" position (toward back of phone) the "Do Not Disturb" feature is ON.

No, that is the Mute switch.