PDOはMySQLデータベースに単純な配列を挿入します

簡単な解決策：

$sql = "INSERT INTO questions (`question_name`) VALUES (:question_name)";
// prepare a stamemnt only once
$stmt = $db->prepare($sql);
$stmt->bindParam(':question_name', $question_name);
// iterate over your POST[question_name] array
foreach ($_POST['question_name'] as $question_name) {
    $stmt->execute();
}