It works for me, maybe you don't have libxml installed with PHP? The blank screen could be the result of a fatal error.
Try changing your script to:
<?php
error_reporting(E_ALL); // log all errors
ini_set('display_errors', 'on'); // display errors in output, rather than to error_log
$url = "http://www.zillow.com/webservice/GetDemographics.htm?zws-id=<my api key>&state=CO&city=Denver&neighborhood=Stapleton";
$data = simplexml_load_string(file_get_contents($url));
print_r($data);
The first two lines will likely result in some sort of error message being displayed which would be a blank page if display_errors is off.