There have been many posts over the past few months about using do_shortcode() with this plugin. The developer has been MIA for about a year, so I found a workaround.
1) Put the shortcode in the content of your page/post: [wp-highrise-contact]
2) Add this code where you want the form to appear:
<?php if (have_posts()) : while (have_posts()) : the_post(); the_content(); endwhile; else: endif; ?>
After taking a quick look at the author's code, I think the problem is with when parts of the code are called with filters, but I could be wrong. Regardless, putting only the shortcode in your page/post then outputting the content works just fine.
I hope this helps everyone who is trying to get this to work with do_shortcode!