How do I hide the shadow under react-navigation headers?

Add the following to the navigationOptions header style.

const AppNavigation = StackNavigator(
  {
    'The First Screen!': { screen: FirstScreen },
  },
  {
    navigationOptions: {
      header: {
        style: {
          elevation: 0, // remove shadow on Android
          shadowOpacity: 0, // remove shadow on iOS
        },
      },
    },
  },
);

The documentation isn’t great yet, but you can learn about navigationOptions in the React Navigation Docs.