The following should work, but isn’t tested:
<div id="header-announcements">
<h3>Announcements</h3>
<?php
$queryObject = new WP_Query( 'post_type=announcements&posts_per_page=5' );
// The Loop!
if ($queryObject->have_posts()) {
?>
<ul>
<?php
while ($queryObject->have_posts()) {
$queryObject->the_post();
?>
<li><a href="https://wordpress.stackexchange.com/questions/26482/<?php the_permalink(); ?>"><?php the_title(); ?></a></li>
<?php
}
?>
</ul>
<div><a href="#">View More</a></div>
<?php
}
?>
</div>