Could it be advisable to go on a totally fat free diet?

No: You need at least 3-5% of your fat as long chained triglycerides to absorb the fat soluble vitamins a, d, e, and k. Without absorbing these vitamins you can get vitamin deficiencies and night blindness ( vitamin a deficiency), rickets, from ( vitamin d deficiency) and irreversible neurological deficiency from lack of vitamin e and bruising and bleeding from vitamin k deficiency.