with this code, I supposed that you would put your logo inside the theme's image folder. This way, the url to image would be:
YOUR-THEME-IMG-FOLDER = probably something like 'images'
IMAGE-FILE.jpg = bionic_icon_white.png
instead, if the full path to image is:
the code should be:
<h1><a href="<?php bloginfo('url'); ?>"><img src="http://bionicpreacher.com/theme_content/bionic_icon_white.png" alt="<?php bloginfo('description'); ?>" title="<?php bloginfo('name'); ?>" /></a><?php bloginfo('description'); ?></h1>