PHP cannot do this, only a client-side language like JavaScript can. That being said, the jQuery library would allow you to do this really easily with its AJAX functionality.
Index.php
Could be refreshed with the following JavaScript:
$("#scores").load("index.php #scores");
That would load the contents of #score from index again without refreshing the entire page.
You could even automate it to refresh every 30 seconds using setInterval();
var $scores = $("#scores");
setInterval(function () {
$scores.load("index.php #scores");
}, 30000);
You can read more about $.load() at http://api.jquery.com/load/#loading-page-fragments.