For me it felt like the immune system got stronger if anything over the course of my 7 day fast. When fat is oxidized and peroxidized the blood becomes a toxic environment for pathogens, I’m guessing. When I broke my fast and ate, I had some mucus buildup so I’m guessing the lymphatic system roared back to life.
So I would say different components of the immune system are affected differently by fasting, and that it also depends on how much fat you’re burning.