Is it dangerous to my health to leave a diseased nail untreated?

Depends. I am not sure what you mean by diseased, but you must mean a fungal infection of the nail. I am of the belief that any infection on your body is not healthy. I may be good to see a podiatrist on this and get his opinion. Since the nail grows so slow, medications have to be taken for a long time to treat the problem.
Could cause problems. Diseased nail could mean infection due to bacterial or a fungus. Abnormal nail growth may lead to ingrown nails. Leaving an ingrown nail untreated could lead to the development of an abscess and possible underlying bone infection. Get it taken care of by a professional to avoid preventable complications.