How do I turn off Featured Content?

If you want to use your own modified Featured Content class, you can take a look at how Twenty Fourteen sets up its own Featured Content instead of Jetpack’s:
https://core.trac.www.ads-software.com/browser/tags/4.3.1/src/wp-content/themes/twentyfourteen/functions.php#L511
https://core.trac.www.ads-software.com/browser/tags/4.3.1/src/wp-content/themes/twentyfourteen/featured-content.php

Jetpack actually looks for other plugins using that class before to declare it, so that should help too:
https://github.com/Automattic/jetpack/blob/3.7.2/modules/theme-tools/featured-content.php#L3

You could use this filter to completely remove the Featured Content file from Jetpack:
https://github.com/Automattic/jetpack/blob/3.7.2/modules/module-extras.php#L23

I hope this helps.

function my_callback_remove_jetpack_modules($tools){
$tools = array(
	'theme-tools/social-links.php',
	'holiday-snow.php', // Happy Holidays!!!
	'theme-tools/random-redirect.php',
	'theme-tools/infinite-scroll.php',
	'theme-tools/responsive-videos.php',
	'theme-tools/site-logo.php',
	'theme-tools/site-breadcrumbs.php',
	'custom-post-types/comics.php',
	'custom-post-types/testimonial.php',
	'custom-post-types/nova.php',
	'theme-tools.php',
);
return $tools;
}
add_filter( 'jetpack-tools-to-include', 'my_callback_remove_jetpack_modules',9,1);

Tried using the above code in my functions.php, with the featured-content.php removed from the tools array. But this does not work.

What could be going wrong?

Okay, I put this code in the site specific plugin and it worked. I reckon this is because the plugins are loaded before the functions.php.

The same can be said about initializing your own Featured Content class.

But Twenty Twelve seems to initialize it in the functions.php itself. Could you please explain how that works, in spite of having Jetpack installed?

I reckon this is because the plugins are loaded before the functions.php.

That’s correct.

Twenty Twelve seems to initialize it in the functions.php itself. Could you please explain how that works, in spite of having Jetpack installed?

Jetpack includes another check within the class itself, right here:
https://github.com/Automattic/jetpack/blob/3.7.2/modules/theme-tools/featured-content.php#L60

If the theme doesn’t declare support, Jetpack bails early.