ASP.NET
Since the question is presented as language agnostic, you might be interested in ASP.NET's Server.Transfer which does exactly what you want to achieve.
PHP
However for PHP, the situation doesn't seem to be very easy and all solutions that come into my mind have design smells:
Sessions
Using sessions to mark your data with certain flags and then check and use them on overview page. You just have to be sure you always unset these data after you don't need them anymore, which might be tricky.
Database
Queue those data in database as confusedGeek described in his post, but I don't think it's a good idea to query every single request like this. It's going to be quite many requests on DB server if you application is bit bigger, which might slow things down.
cURL
Taking advantage of cURL in PHP, if you have the chance:
<?php
$curl = curl_init( );
curl_setopt( $curl, CURLOPT_URL, "http://localhost/something.php" );
curl_setopt( $curl, CURLOPT_POST, true );
curl_setopt( $curl, CURLOPT_POSTFIELDS, $_POST );
curl_exec( $curl );
curl_close( $curl );
?>
<form action="test.php" method="post">
<input id="textfield" name="textfield" />
<input type="submit" />
</form>
This piece of code takes something.php on server side, allows to send POST data to it and shows you its content (which could be just print_r( $POST ); in this example). This one could do what you need, but it has once again one flaw - the URL won't change, so users might get confused - and I wouldn't really recommend it.
I personally think your case might be a design flaw. Why would you want to take all resources from form page and move them to other? Isn't easier to work with your data in a file / class designed for it and then just decide what outcome you have? If something fails, it returns the user on page with form and if everything went well, you post the data to DB and show overview page with some happy message that everything went okay.
If you really want to proceed with the way of sending everything to other page, using AJAX/AJAJ could be another solution for you.