Ok, so something in your custom theme/plugin is removing the icons from the DOM. If you inspect the page source of any Discourse site, you will see this towards the end of the page:
That class="fontawesome" element contains the icons. In your site, something removes that element and replaces it with some other stuff. Here’s a screenshot from your site’s page source:
You need to figure out what specifically does this, and modify it so it doesn’t wipe out the icons.