Reddit reviews Missing Microbes: How the Overuse of Antibiotics Is Fueling Our Modern Plagues

As far as I know, there hasn't been much experimental work examining the consequence of hand-washing only on the development of autoimmune disease or immune dysregulation. But there is a significant amount of evidence that the Hygiene Hypothesis is correct in at least some important respects. It is too much to describe in detail here but we can review a few examples to give you a sense of why immunologists generally support the idea.

The incidence of autoimmune disease and helminth exposure is inversely correlated -- people in parts of the world where exposure to parasitic worms is common rarely develop autoimmune diseases to the extent that those in developed, ultra-hygienic countries do. (The latter also happen to be the places on Earth where the highest amounts of antibiotics are used.) What's more, in mouse models of autoimmune disease, such as NOD (type I diabetes) and EAE (multiple sclerosis), disease severity is reliably reduced when helminth therapy (e.g., immunization with S. mansoni eggs) is administered. This suggests that the relationship is not merely correlative and may be causal. Induction of the anti-inflammatory cytokines TGF-β and IL-10 are thought to mediate this effect.

In addition to secreting immunomodulating compounds that bias the host immune response to promote tolerance of their presence, certain commensal bacteria, archaea, and fungi, as well as parasitic worms, induce the differentiation of T regulatory cells (Tregs) that dampen inflammatory responses generally. Early exposure to diverse microbial populations is associated with decreased iNKT cell numbers and less inflammation, which is likely why offspring born by Cesarian exhibit a higher incidence of immune disorders than those exposed to vaginal flora during natural birth.

Thus, broadly speaking, there is substantial evidence to support the Hygiene Hypothesis, that early exposure to microbes "programs" or "tunes" the immune system to function appropriately, and that dysbiosis produced by frequent washing with detergents and decreased exposure to commensal organisms we've evolved for hundreds of thousands of years with, contributes to impaired and/or inappropriate immune responses.

Further reading: It's been a few years since I read it but I found Jessica Snyder Sachs' book Good Germs, Bad Germs to be an accurate, accessible introduction to the role of commensal microbes (mostly focused on bacteria) in maintaining immune function -- perhaps you will find it interesting. I have not read Martin Blaser's Missing Microbes but it also looks like a valuable general audience book on similar topics.