What are foods that one should avoid if one has an autoimmune disease such as arthritis, lupus or fibromyalgia?

Probably none: While many diets are promoted as therapy, there are no controlled studies to support any benefit.

Active investigation: This is an excellent question, and i agree with my colleagues that the answers are far from clear. However, some foods may be more pro-inflammatory than others, and may induce immune response. For example, wheat & cereal grains: http://www.Ncbi.Nlm.Nih.Gov/pubmed/23482055 this may not apply to everyone, but some people are sensitive to these products and may improve if eliminating them.