Since it is a GET request, you don't need this line which is for post:
curl_setopt($curl, CURLOPT_POST, true);
Furthermore, you are authenticating incorrectly. The authentication stuff should be in the headers, not in the body of the request.
Instead of
$curl_post_data = array(
"X-Username" => 'EMAIL ADDRESS',
"X-Api-Key" => 'API KEY',
);
curl_setopt($curl, CURLOPT_POSTFIELDS, $curl_post_data);
you should have something like:
curl_setopt($ch, CURLOPT_HTTPHEADER, array(
"X-Username" => 'EMAIL ADDRESS',
"X-Api-Key" => 'API KEY',
));
Note, I don't have experience with Tierion's data api. I just read the documentation you linked to, I highly suggest that you give that a read and fully understand how the API works, and how HTTP Requests work.