Is a person sinful if they gave a diabetic patient a second dose of insulin, which led to their death, knowing that they asked the conscious patient if he had taken the dose and he denied it, and it was later discovered that another neighbor had given him a previous dose?

There is no sin upon the questioner for giving the injection to his elderly neighbor. Rather, he is thanked and rewarded for his care and attention towards him. He has done what is permissible for him to do, as he asked his neighbor and received permission. Permission negates liability. If someone performs a permissible act with permission, and death or damage results from it, there is no liability upon him. This is what the jurists have unanimously agreed upon. Furthermore, the death of the neighbor is merely a supposition to which no ruling is attached, and the default is innocence of liability.