Hi there, no you don't require a dedicated server. Here's an extract from the FAQ page:
AppThemes does not work as a standalone solution. It is a theme which runs on the free CMS blogging software, WordPress which requires a web server such as apache, a mySQL database, and PHP. If your hosting provider allows WordPress to run on it (over 95% of them do), then AppThemes should work fine. You might also need FTP access to upload the theme and admin permissions to change/create folders. This is very common for any web hosting provider. We are partners with some of the best affordable and reliable web hosting providers out there today. They are certified WordPress partners and will have no problem running our themes.
