<label for="inp">Inserire codice fiscale</label>
<input style="width: 100%;" id="inp" name="inp" type="text" />
<div id="yeah" style="display: none;">show some text</div>
<div id="no" style="display: none;">show another text</div>

$(function(){
  $('input[name=inp]').keyup(function(){
    var inpvalue= $('#inp').val();
    $.ajax({
      type: 'POST',
      data: ({p : inpval}),
      url: 'check.php',
      dataType: 'json',
      success: function(data) {
        if(data.status == 'success'){
          $('#yeah').show();
          $('#no').hide();
        }else{
          $('#yeah').hide();
          $('#no').show();
        }
      }
    });
  });
});

<?
function add_my_script() {
wp_register_script(
        'cf',
        get_stylesheet_directory_uri() . '/js/cf.js',
        array('jquery')
    );

    wp_enqueue_script('cf');
}

add_action( 'wp_enqueue_scripts', 'add_my_script' );

?>

<?php
    $dbhost = 'localhost';
    $dbuser = 'user';
    $dbpass = 'pass';
    $conn = mysql_connect($dbhost, $dbuser, $dbpass);


    if(! $conn ) {
        die('Could not connect: ' . mysql_error());
            }

$cf = $_POST['p'];
string strtoupper (string $cf);

$sql1 = "SELECT cf FROM table WHERE cf='$cf'";

   mysql_select_db('database');
   $retval1 = mysql_query( $sql1, $conn );

   $count1 =0;


   if(!$retval1) {
     die('Could not enter data: ' . mysql_error());
   }

    while($row = mysql_fetch_array($retval1, MYSQL_ASSOC)) {
    $count1 = $count1 + 1;
   }

    mysql_close($conn);

    if($count1 > 0){
        $response_array['status'] = 'success';
            header('Content-type: application/json');
    echo json_encode($response_array);
    }

?>