I’m not sure what the issue is but for me, I can see the house icon on both localhost and your live website. The GitLab icon in the footer is missing on the live site though.
I’m not familiar with building on gitlab, but my guess is the mistake is somewhere in either how the extended version of hugo works with gitlab pages, or with what you’re pushing into your git repository.
yeah, I just looked at the inspector in Firefox too, and I don7t see anything in the head that loads the font awesome library. I see an icon with some small characters in it, but, not the bars icon that this is calling: