I loaded your test site and immediately noticed a couple of issues. Not 100% sure what exactly is causing the disappearing header, but here are some things to check:
style-options.php
Your main stylesheet is using @import url("functions/stylesheets/style-options.php"); to import a stylesheet. My guess is that it’s trying to import some dynamically-generated css. Unfortunately, this is what that stylesheet contains:
Deprecated: Function set_magic_quotes_runtime() is deprecated in /home/deb28386n2/domains/jethrowaanders.nl/public_html/test/wp/wp-settings.php on line 32
Warning: Cannot modify header information – headers already sent by (output started at /home/deb28386n2/domains/jethrowaanders.nl/public_html/test/wp/wp-settings.php:32) in /home/deb28386n2/domains/jethrowaanders.nl/public_html/test/wp/wp-content/themes/paperpunch/functions/stylesheets/style-options.php on line 4
It’s also returning with a content-type of text/html, which IE9 is rejecting altogether as a type mismatch (stylesheets should only be delivered as text/css).
Cufon
Your theme is trying to use a system called Cufon to generate the header and navigation link text. In modern browsers (not IE), it uses HTML5 canvas to do this. In IE it switches over to VML (vector markup language).
IE9 should still be using canvas, though. The demo theme from ThemeFoundry is still using canvas on the header … yours is attempting to use VML. Why it’s switching is likely a change or configuration you made somewhere while customizing the theme.
There are also a couple of other brute-force fixes you can use if you can’t narrow down exactly what you changed. Here’s one.