Font awesome icons are not actual images, so this might be a tough one. Category images are not supported in wordpress either, but that can be resolved with a plugin like categories images – the search integrates with that one.
The other option is to do it programmatically, but displaying the font awesome images requires changing the results templates as well, to accomodate the new class names etc.. I would only recommend that if you have good experience in custom coding, or have a developer doing it for you.
Best, Ernest Marcinko If you like my products, don't forget to rate them on codecanyon :)