If the goal of your fast is to burn body fat, YES! But it should be black coffee (no sugar, artificial sweetener, or milk additives) and the fresher the roast the better!
Coffee, and caffeine, encourages fatty acids to get mobilized, go to the liver and get converted into Ketones. This gives you more energy from not only the caffeine, but from the more Ketones. This state, referred to as Ketosis, is when your body converts fat to glycogen which can then be used as energy.
Another benefit is that the polyphenols in coffee allow a greater autophagy effect. Autophagy is the bodies way of recycling old dying cells, and one of the main benefits of Intermittent Fasting. For more information on this topic research Intermittent Fasting, Dr. Eric Berg, Dr. Jason Fung, and Thomas Delauer.
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