I tested the code and it works. Any questions just ask.
index.php
!DOCTYPE html>
<html>
<head>
<title></title>
<script src="//ajax.googleapis.com/ajax/libs/jquery/1.11.2/jquery.min.js"></script>
<script type="text/javascript">
function atualizarTarefas() {
// aqui voce passa o id do usuario
var url="get.php?id=1";
jQuery("#tarefas").load(url);
}
setInterval("atualizarTarefas()", 1000);
</script>
</head>
<body>
INFORMACAO EH EXIBIDA AQUI: <div id="tarefas"></div>
</body>
</html>
get.php
<?php
$con = mysqli_connect('localhost', 'root', '', 'tarefas');
$id = $_GET['id'];
$sql = "select * from tarefas where id = $id";
$query = mysqli_query($con, $sql);
$resultado = mysqli_fetch_assoc($query);
$retorna = $resultado['vencidas'];
echo $retorna;
?>