How to hide first gallery for every post?

Question

You could use preg_replace() but I think it’s much easier to overwrite the $output via the post_gallery filter and then remove the filter after the first run:

/**
 * Remove the output of the first gallery
 *
 * @param string $output
 * @param array $attr
 * @return string $output
 */
function wpse125903_remove_the_first_gallery( $output, $attr )
{
    // Run only once
    remove_filter( current_filter(), __FUNCTION__ );

    // Override the first gallery output        
    return '<!-- gallery 1 was here -->';   // Must be non-empty.
}

add_filter( 'post_gallery', 'wpse125903_remove_the_first_gallery', 10, 2 );

This should remove only the first gallery in your posts.