PHP variable as form action

No.

PHP is executed on the Server. When you post this form PHP only know content of that variable when post is there. So on first-rendering the page the variable of course is empty.

But you could do this with Javascript:

<form id="postForm" method="POST" action="">
    <label>
        Post to:
        <select project="location" id="location" name="location">
            <option value="first"></option>
            <option value="handle-something">Option1</option>
        </select>
    </label>
    <input id="button1" type="submit" name="submit" value="Post">
</form>

<script>
    document.getElementById('location').onchange = function() {
        // maybe you can get val via this or so
        var location = document.getElementById('location').value;
        document.getElementById('postForm').action = '/' + value + '.php';
    };
</script>

Not testet.

While you can generate the form dynamically on render, you cannot change it with php after the form is served. You will need to use javascript for that as @copynpaste already shows. There is another option you can use.

Send the form to a standard action and extract your variable on the server-side where you decide what to do.

example:

<form method="POST" action="formaction.php">
        <label>Post to: <select project="location" id="location" name="location">
            <option value="null"></option>
            <option value="$var1">Option1</option>
            ...
        </select><br>
        <input id ="button1" type="submit" name="submit" value="Post">
</form>

Inside formaction.php

$action = $_POST['action'];

include "{$action}.php";

This will include a file based on the value of "location".

Since it's unlikely that you would want to allow post data to go to a potentially arbitrary, uncontrolled file location based upon unsanitized post data, I like the idea of examining the data's values for acceptable values; then applying a 'default' catch-all to address two cases: (1) a situation where the form is loaded without post data; and (2) a situation where the form is loaded with unanticipated or unacceptable post data:

<?php 
    $location = $_POST["location"];
    switch( $location )
    {
        case 'create':
        case 'remove':
        case 'update':
        case 'delete':
            $form_location = $location;
            break;
        default:
            $form_location = 'default';
            break;
    }
?>
<form method="POST" action="../<?php echo $form_location; ?>.php">
    <label>Post to: <select project="location" id="location" name="location">
    <option value="null"></option>
    <option value="$var1">Option1</option>
    ...
    </select><br>
    <input id ="button1" type="submit" name="submit" value="Post">
</form>