How to change a user’s role depending on date registered?

Close, your date comparison is slightly off – user_registered will be a MySQL datetime, so your string comparison will never evaluate true. Instead, convert it to just the date:

$registered = mysql2date( 'Y-m-d', $user->user_registered ); 

Now we’re talking:

if ( '2015-10-01' === $registered ) {
    // chocks away
}

Update: Here’s a complete snippet that should be less intensive on memory too:

$users = get_users(
    array(
        'fields' => array( 'ID', 'user_registered' ),
        'role'   => 'subscriber',
    )
);

foreach ( $users as $user ) {
    if ( '2015-10-01' === mysql2date( 'Y-m-d', $user->user_registered ) ) {
        $u = new WP_User( $user->ID );

        // Remove role
        $u->remove_role( 'subscriber' );

        // Add role
        $u->add_role( 'to_be_approved' );
    }
}

Code is not tested and you should assume it will be break, so backup, backup, backup. Make sure you have debugging enabled too.