As a rule of thumb, fats and animal proteins of any kind should stay out of your home compost pile. Essentially, most home piles do not get hot enough to break these materials down, creating the potential for rot in your pile. Oils also make it hard for oxygen to get through, which means your pile will probably become anaerobic rather than aerobic. While anaerobic home compost will still break down eventually, it takes longer to do so and smells worse. Additionally, cooking oils and grease in quantity can attract animals to your pile.
Stick to non-greasy, non-processed foods instead for the best results in your compost!