Do dogs lose their fur if they get chemotherapy?

No. While veterinarians will treat dogs with cancer using chemotherapy, in general veterinarians do not allow dogs to have the side effects of chemotherapy that are tolerated in human medicine. If a dog starts to vomit, has diarrhea, becomes lethargy or withdrawn, the chemotherapy is reduced or stopped to preserve the dog's quality of life.

I've seen over a hundred dogs in chemotherapy and none have had any chemotherapy-related fur loss.