This isn't making sense. FrmEntry::create( $values ) doesn't use Ajax. It's a PHP function that runs on the server. There's no AJAX involved unless you're triggering Ajax from the front end. Your screen capture of your Ajax just appears to be setting a cookie. There's no redirect being set. Regardless of what you are trying to do, the jQuery you are using to set a cookie via Ajax will always return an Ajax response. FrmEntry::create( $values ) returns an id for the newly created entry, which you are not capturing on a variable like this: $newid = FrmEntry::create( $values ). There's not enough information about your process provided to understand what's going wrong. Where is the FrmEntry::create( $values ) complete code running? What filter or function are you using.

Thanks for taking a look @Victor Font! The AJAX script in the screenshot is being generated by Formidable when I call FrmEntry::create().  You can see their code in formidable-pro/classes/views/frmpro-entries/set_cookie.php:

<?php
if ( ! defined( 'ABSPATH' ) ) {
	die( 'You are not allowed to call this page directly.' );
}
?>
<script type="text/javascript">
jQuery(document).ready(function($){
jQuery.ajax({type:'POST',url:'<?php echo esc_url_raw( admin_url( 'admin-ajax.php' ) ); ?>',
data:"action=frm_entries_ajax_set_cookie&entry_id=<?php echo (int) $entry_id; ?>&form_id=<?php echo (int) $form_id; ?>"
});
});
</script>

  In this example:

FrmEntry::create( $very_cool_data );
header(\'Location: VERY COOL URL\' );
echo \"did not redirect...\";

The header call silently fails and the output to the page is what\'s in my screenshot. Again, the screenshot is what is generated by Formidable when FrmEntry::create is called, not something we wrote.

Really, all I need to do it manually create an entry at the top of a template file and then redirect to another page. We\'ve done this in the past without issue, so I was hoping someone could help explain why FrmEntry::create() generates HTML output.

If anyone else finds themself here, I believe I have figured out a solution. Digging through the code base, there is a check for a POST parameter called frm_skip_cookie. See below. 

public static function maybe_set_cookie( $entry_id, $form_id ) {
        if ( defined('WP_IMPORTING') || defined('DOING_AJAX') || defined('REST_REQUEST') ) {
            return;
        }

        if ( isset($_POST) && isset($_POST['frm_skip_cookie']) ) {
            self::set_cookie($entry_id, $form_id);
            return;
        }

        include(FrmProAppHelper::plugin_path() . '/classes/views/frmpro-entries/set_cookie.php');
    }

In my code, I added this to the data being sent to FrmEntry::create():

            $entry = array(
                'form_id' => $form_id,
                'item_key' => $item_key
                'frm_skip_cookie' => true,
                'item_meta' => $data
            );

            $_POST = $entry;
            $entry_id = FrmEntry::create( $_POST );

            header('Location: '. $redirect_url );

This now creates the entry and generates no HTML output.