Removing Facebook contact field from user contact not working

Firs things first… always, always enable debugging. Secondly, you should accept the $user parameter in your callback – otherwise you are always checking the logged in user, not the user within the current context:

function modify_user_contact_methods( $methods, $user ) {
    if ( ! $user instanceof WP_User ) {
        if ( ! $user = wp_get_current_user() )
            return $methods;
    }

    if ( strtotime( $user->user_registered ) > strtotime( '1 year ago' ) ) {
         unset( $methods['facebook'] );
    }

    return $methods;
}

add_filter( 'user_contactmethods', 'modify_user_contact_methods', 10, 2 );