AirPlay Mirroring games to AppleTV?
If your hardware supports it, it will work.  My mid-2010 mini doesn't, but my son's mid-2011 mini does support it.  He plays games on the Apple TV + big-screen TV.  He even brings his Bluetooth keyboard and mouse into the TV room, and they're close enough to the Mac mini to work.

Performance is NOT great on the high-framerate FPS kind of games, but should be good for Minecraft.

Even when your HW does not support the built-in mirroring, there are some third-party SW that enables it.  Try AirParrot, Beamer, and Bruce (no affiliation).