You can get fat eating too much of anything, even officially healthy foods.
A fat tax would lead to similar smuggling of food the way cigarettes are smuggled into areas with high cigarette taxes.
You'll see people taking steps around any rationing and portion limits, such as people switching to buying more 2 liter sodas when single serve sodas were taxed and limited in New York. Or there will simply be organized shopping rings to get around it like having your kids buy items in cash ahead of you so that you buy more without being restricted to X when Mom pays.
There's no consensus on classes of food being bad on you, such as butter alternatively being good and bad. A fat tax therefore won't help - and you'll get outcries from people on Atkins or disagree with official consensus today that the food is bad. So it can't stand