This is not theme related issue but your hosting/setup related issue as theme doesn’t create any redirects. Font Awesome library will work either using example.com/blog or blog.example.com because theme uses relative URLs for linking static content and it doesn’t matter what URL you are using.
I am not sure of your existing setup but you might have to do search and replace on your database to replace old URL with new URL or just replace site URL inside WordPress Dashboard - General. Either way some other things shouldn’t work either if URLs are the ones to blame.
Do you have a live site where I can see what is going on with Font Awesome icons?
This is the rewrite rule I have in my htaccess, because I changed the old links.
RewriteRule ^([0-9]{4})/([0-9]{2})/([0-9]{2})/(.*)$ http://www.blog.speedcomputers.biz/$4 [L,R=301]
While your main URL is set to blog.speedcomputers.biz it still tries to load everything from speedcomputers.biz/blog and things gets broken. If you want to load your website from blog.speedcomputers.biz then you should replace it everyhwere to make things constant everywhere. Or at least hardcode URLs for fonts from blog.speedcomputers.biz. Another option is to load Font Awesome library from CDN. Here you can find one CDN option for Font Awesome fonts.
Probably you can find some other workaround as well.