Why does is_user_logged_in() return false after I change user password on the front end?

The login cookie is named 'wp-postpass_' . COOKIEHASH where COOKIEHASH is either defined in your wp-config.php or in wp-includes/default-constants.php function wp_cookie_constants().

The value is:

$hasher->HashPassword( wp_unslash( $_POST['post_password'] ) )

Whenever the password is changed, the cookie doesn’t match anymore, and is_user_logged_in() must return FALSE.

Send your users to the login screen immediately after they changed the password.
Or authenticate the user after the password saving:

require_once ABSPATH . 'wp-includes/class-phpass.php';
$hasher = new PasswordHash( 8, true );
$expire = apply_filters( 'post_password_expires', time() + 10 * DAY_IN_SECONDS );
setcookie( 'wp-postpass_' . COOKIEHASH, $hasher->HashPassword( wp_unslash( $_POST['post_password'] ) ), $expire, COOKIEPATH );