Losing fat itself is not harmful for hair at all, but the process to losing fat could be dangerous. A lot of people tend to eat way less than usual when they are on diet, which results in lack of nutrition. Since scalps are starved of necessary nutrition, they fall out as a result. Losing some fat in a long period definitely does not cause hair loss. Crash or rapid weight loss on the other hand can be detrimental for hair loss. My hair loss began from one and I've seen many people who lost hair from going on severe diet.