Properly disposing of dead animals is important for many reasons, including your health, wealth and the well-being of surrounding animals. In addition, when an animal is properly disposed of citizens who don't hunt have less to critique since they aren't bothered by the sight or smell of the carcass. Since hunters tend to be in the minority of any given state's population, it's crucial to keep up positive relationships with other citizens.
Your health is a major factor when calculating the importance of proper disposal of dead animals. You don't want the animal's remains to spread any diseases to you through direct contact, nor do you want the remains to leach out diseases through the ground. That's why you want to wear gloves when handling the animal directly, and once you choose a method of disposal, make sure it won't leach out diseases into the local water source.
Your wealth is another factor to take seriously when you are weighing the best disposal method. Though some sanctioned ways of getting rid of dead animals do cost money, it's in the best interest of your wallet to use those rather than dumping the remains illegally. Such unacceptable methods would result in a fine if you were caught by authorities.
Since you appreciate the natural world, it's important to consider the well-being of living animals when you are choosing a method of disposal. For instance, if you bury an animal, make sure vultures can't get to it, especially if the animal was at risk for a disease. You wouldn't want to infect any innocent animals.
Now you're equipped with the basic knowledge of what methods are acceptable when disposing of a dead animal and the importance of properly using those methods. Happy hunting!