According to an Interview conducted by Mashable with Shigeru Miyamoto, you won’t be able to play Super Mario Run offline. You always have to be connected to the internet to play the game.
Here’s a section of the article:
Mashable: “I learned today that Super Mario Run requires an internet connection to play. What’s the reason for that? Are there any thoughts about an offline mode?”
Shigeru 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.
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. This is something that we want to continue to work on as we continue to develop the game.
But actually, the security element is one of the reasons that we decided to go with iPhone and iOS first. So this is just — based on the current development environment — a requirement that’s been built into the game to support security and the fact that the three different modes are connecting to the network and interacting with one another.
We had thought at one point that it would be nice to have the World Tour [story] mode available standalone, to be able to play without that connection. But then the challenge is when that’s operating in a standalone mode, it actually complicates the connection back to the Toad Rally and Kingdom modes. And because those two modes are relying on the network save, we had to integrate the World Tour mode as well.”