How to Disable Color Options?

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In TwentyTen’s functions.php the custom background is added in twentyten_setup() which is called on the action 'after_setup_theme'.

So, in your child theme, the following should remove the option completely:

// Higher priority to work after TwentyTen
add_action( 'after_setup_theme', 'wpse12934_remove_custom_background', 20 );

function wpse12934_remove_custom_background()
{
    remove_custom_background();
}