It looks like the cart is from a plugin, so you have to account for the change in directories.
The button should probably look like:
<input type="button" name="eCart1_Continue_100" id="eCart1_Continue_100" value="Continue Shopping" class="eC_FormButton" onclick="document.locaton.href='../../product_results.php'" />
Just make sure the path to the product_results.php page is the correct path relative to the page you are updating.