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.
This is first time, I'm working on API and its really painful. I'd really appreciate if you can write code to authenticate. I'm sure, I can write GET, POST command thereafter.