Super Mario Run cannot be played offline as players will need to be connected to the internet at all times just to play on it.
Shigeru Miyamoto confirmed this with an interview with Mashable a few days ago. The reasoning is a number of things. Firstly, piracy on iPhone games is an issue so the online DRM prevents people from hacking the game.
Secondly, Nintendo wants all of the game modes in Super Mario Run to operate smoothly. Many mobile video games require an always-online connection, Sadly though, this means you cannot play Mario on an airplane.
Don’t expect Nintendo is forcing online DRM to all of its games. This is only tied to Super Mario Run on mobile. It’s unlikely Nintendo will do this with Nintendo Switch games as that console’s premise is that you can play games anywhere you want, whenever you want.