I am having the same problem. Notifications have all been turned completely off in settings for BOTH game center and "Deer Hunter 2014" game. Banners even set to "none".
I even tried deleting the game from my Game Center games, but it then asks me if I want to uninstall the game from the device, which I do not because I would lose a lot of progress.
BUT THE NOTIFICATIONS KEEP POPPING UP.
This is particularly irritating and obnoxious because it actually covers up critical controls in the game at the top of the screen. As though knowing that "you have achieved *I see you* by killing 100 animals with infrared" is more important to me than actually playing the game. IT'S NOT. I DON'T CARE ABOUT GAME CENTER. IT'S A BAD IPHONE FEATURE. And Apple is way too proud of it and it's cute little colorful bubbles.
Please send help and fix the things you've done.