Non-alcoholic fatty liver disease most commonly occurs in people with overweight people with insulin resistance- in type 2 diabetics, in fact. It's fairly treatable with a low fat diet.
Roughly 40% of type 2 diabetics have NAFLD.
I expected, when I looked for prevalence studies of NAFLD in type 1 diabetics, to find that we'd have lower rates- near normal rates. We don't. I found only two prevalence studies of NAFLD in type 1 diabetics (both on adults) and they gave prevalences of 44% and 53%
There are other liver diseases that are linked strongly with diabetes. One of the more unusual ones is called glycogenic hepatopathy, which is an acute liver disease caused by high blood sugars over a medium term- not a long term complication but you don't develop it in a week either (the cases I've read about have been in people with type 1 diabetes and A1cs between 10% and 14%). It goes away when you get better blood sugar control. It can be differentiated from NAFLD by liver biopsy. Glycogenic hepatopathy is one of those complications of diabetes that I figure I really do have the power to stop myself from getting- not maybe, but definitely.