The body can only absorb vitamins (and many minerals) in the presence of fats. So a totally fat free diet would rapidly lead to illness (and, eventually, death) through vitamin deficiency.
Unless the idea of rickets, scurvy, etc, really appeal to you, keep eating some fats !!
Have a look at Rosemary Conley's Hip and Thigh Diet, because that is virtually fat free. She had to go on a totally non-fat diet in order to avoid a gall bladder op some years ago, which is how she discovered the money-making idea (although having known the lady personally I can honestly say she deserves it).
The diet can be done and it does work - I lost two stone in six months.