I've poked around in the menu a bit, but I don't see a way to turn off the tutorial for Fallout Shelter. Having lost a vault to a bugged bug (an unkillable radroach) I'd prefer to not have to go through the whole tutorial again.
According to Bethesda's official Fallout Shelter Updates thread, the ability to skip the tutorial was added in version 1.2:
Are you an experienced Overseer who has already undergone Vault operational training? Now you can skip the tutorial!
When you start a new vault, there is an intro screen congratulating you on being selected as the new overseer. In the upper-right corner of this screen you will find the option to 'skip'. This skips the tutorial.
Here is the screen I'm referring to, pre-patch:
The game will start with the vault entrance, three levels of elevator, and a line of dwellers outside the vault.