Can I turn off the Tutorial in Fallout Shelter

There is no way to stop or skip the tutorial. You must play it through, however you do not need to put the rooms in the spots the arrows are pointing, and can put them anywhere you would normally be able to put them.

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.