So, there is some complicated science behind it, and that science is important to both Type 1 and Type 2 diabetics, but not necessarily in the way people peddling diets might have you believe. From what I can tell (from perusing the peer-reviewed literature on the subject), the two most important factors in diet that contribute to inflammation are levels of insulin and consumption of Omega-3 and Omega-6 fatty acids.
In particular, arachidonic acid (an Omega-6 that is either ingested or converted from linoleic Omega-6) is the main, known culprit in inflammatory responses linked to chronic disease (including metabolic syndrome and insulin resistance). Ingesting arachidonic acid (which is common in some Omega-6 rich oils) seems pretty negative if someone has the genetic profile to be prone to inflammatory responses to diet; linoleic acid is contraindicated in those who have high serum insulin levels and a genetic predisposition to inflammatory response. So, the general thinking based on clinical studies is pretty straightforward: decrease carbohydrate consumption and limit or eliminate high GI foods; decrease Omega-6 intake; increase Omega-3 and polyphenol-rich foods.
In practice, that means a diet low in grains, high in fish and flax oil, low in refined “vegetable oils,” low in carbohydrates, and high in colored (red and green) vegetables. A becoming-standard “nutritional ketogenic” diet with heavy Omega-3 supplementation would likely be a pretty successful anti-inflammatory diet. As for the effectiveness of such a diet, I can only find evidence that it (the anti-inflammatory diet) is useful for combating insulin resistance, rheumatoid arthritis, chronic GI tract disorders, and obesity. As a side note, they’re talking pretty serious Omega-3 supplementation: 3g of EPA & DHA per day, which is as far as I know only obtainable (reasonably) from fish oils.
Some references that might be useful if you can get institutional access:
N.B. There is a lot of pseudoscience tomfoolery out there on such diets, which makes sense (there is money to be made in peddling diets). The links I’ve provided above, however, are to peer-reviewed research with direct application to insulin resistance and R.A. As with all dietary research, however, it is difficult to say how generalizable results of such studies are. You may find such a high Omega-3, low carb diet useful, or you may not. YDMV