Note: This post is specific to US policy.
Type 1 Diabetics who are alive are allowed to donate blood if they've never received bovine insulin from the UK and meet other blood donor requirements (I miss the requirements because I'm too small). Diabetics are more likely than the general population to be excluded for answering that that they do not feel healthy, or taking aspirin, or having kidney disease, etc.
Living type 1 diabetics are generally barred from donating anything else. I believe type 2s are as well but the policies I found were somewhat vague as to a person with
well controlled type 2 diabetes not on insulin might be eligible.
People with all autoimmune diseases are barred from donating bone marrow (at least on paper- my mother was accepted for the bone marrow registry and she has rheumatoid arthritis) and people who take medications to control diabetes are also barred.
Dead diabetics, on the other hand, can be organ donors. There has been some discussion in the medical literature about how healthy a dead person has to have been to make a good enough organ donor, particularly in regards to organs for which their is a shortage, such as kidneys. In 1995, the net of potential donors was expanded to accept cadaver donors who had had diabetes or hypertension or who had been dead a little longer.
People waiting for organs in the United States are allowed to indicate whether or not they are willing to accept an organ from an "Expanded Criteria Donor"; such donors would include dead diabetics. The pancreas would likely not be viable, but kidneys (even if you have microalbuminuria), liver, heart, and lungs,
not to mention corneas- these could extend somebodies life.
If we count things you are allowed do donate dead or alive, then just about the only things diabetes stops you from donating are your pancreas (which you might be able to donate to a research study) and your bone marrow.
That means that being diabetic is no excuse not to register as an organ donor.