Enable featured image but not have it show up?

TwentyEleven includes the featured image from header.php, rather than a template part like most themes. If you comment out line 92 that should remove it, e.g.

<?php
    // The header image
    // Check if this is a post or page, if it has a thumbnail, and if it's a big one
    if ( is_singular() &&
            has_post_thumbnail( $post->ID ) &&
            ( /* $src, $width, $height */ $image = wp_get_attachment_image_src( get_post_thumbnail_id( $post->ID ), array( HEADER_IMAGE_WIDTH, HEADER_IMAGE_WIDTH ) ) ) &&
            $image[1] >= HEADER_IMAGE_WIDTH ) :
        // Houston, we have a new header image!
        // echo get_the_post_thumbnail( $post->ID, 'post-thumbnail' );
    else : ?>
    <img src="https://wordpress.stackexchange.com/questions/41991/<?php header_image(); ?>" width="<?php echo HEADER_IMAGE_WIDTH; ?>" height="<?php echo HEADER_IMAGE_HEIGHT; ?>" alt="" />
<?php endif; // end check for featured image or standard header ?>

However, it’d be better to create a child theme and place your modified version of header.php inside it.