Is it possible in iOS 6 to make banners only appear when the device is active, but self-dismiss upon unlock? If I'm using the device, I'd like to see banner notifications from other apps.

When I unlock it, I don't want a swarm of banner notifications bothering me and hindering my ability to use the top of whatever app I'm using. I'd rather just check the notification center at my leisure.

EDIT: But I still want the rest of the device to operate as per usual; in particular, I don't want to turn on Do Not Disturb b/c I want to be disturbed by whatever else I've set to disturb me.

Is this possible?
Yes it is possible.  Turn on Do Not Disturb when you are in your phone.  

Or just turn off notifications 

http://www.imore.com/how-customize-notification-center-alerts-your-iphone-and-ipad