According to Miyamoto, “For us, we view our software as being a very important asset for us. And also for consumers who are purchasing the game, we want to make sure that we’re able to offer it to them in a way that the software is secure, and that they’re able to play it in a stable environment.”
When asked if by “secure” they mean against the risk of piracy, Miyamoto confirmed this. However he also says that this is to keep all three game modes functioning together. “We wanted to be able to leverage that network connection with all three of the [Super Mario Run] modes to keep all of the modes functioning together and offering the game in a way that keeps the software secure.”
We’re not sure why an internet connection would be needed, but unfortunately that’s the case for now. So if you plan on playing the game while you’re in a plane or somewhere without internet, you’d be out of luck. In the meantime Super Mario Run has been set for a release on the 15th of December but a playable demo is already available in Apple’s stores.