Does the way an animals are killed cause illness to humans? It certainly can, both a psychological illness and physical illness with even death! The real question should be how can the way an animal is killed cause illness in humans and are there ways it can be prevented? That is a bit more complex of a question and the answer requires more thought!
Illness can be defined in several ways. For the purposes of this paper it will be defined using the first definition found in American Heritage® Dictionary of the English Language as given by the Free Dictionary which reads "Poor health resulting from disease of body or mind; sickness." It needs to be noted that illness can be mental as well as physical, and the mental can have physical symptoms.
Let us deal with the psychological issues first. Many people love animals and become distressed at the sight of blood. It can cause numerous nervous disorders, loss of sleep and even vomiting and upset stomachs. This is factual and varies from individual to individual. A "humane" killing, such as through sleeping drugs or even electrocution can prevent these reactions, avoiding the associated psychological illnesses. Fortunately this does not effect many people. Can the way an animal is killed cause illness aside form mental problems?
On thing that needs to be decided is does poisoning count as an illness? It can be terminal and kill a human, and certainly weaken the immune system. This can then make the person more susceptible to disease and illness. Poisons are used for killing some animals and can be passed on to anything that eats the animal. Cats and rats are a prime example, but fish can be poisoned and if eaten by humans have adverse effects!
Then there is killing by force, and in some ways this can cause illness in humans. Processed animals must be killed in a fashion that allows them to bleed out. The blood can breakdown and increase the growth of bacteria that can cause illness. This is the reason for the ancient laws found in the Bible, Torah, and Koran. Also, if a hunter kills an animal with a gut shot, it needs to be cleaned thoroughly as the bacteria from the intestine will get on the meat. Extremely good cleaning and then cook well done will prevent illness for this type of killing.
In reply to the question, "Does the way an animals are killed cause illness to humans?". the answer is yes, it can.