Just because a food is labeled as fat free does not mean that it is not healthy for one to eat. In fact the fat free food item may be a better option to pick from instead of something else that may contain a high number of fat that goes over what one should eat.
The only healthy fat free foods are the ones that are naturally fat free. Those that are processed and changed to make them fat free are full of chemicals, fake dyes and flavors, and a whole host of other unhealthy, unnatural, harmful substances. In addition, the body needs some healthy fat to function well, so eating a diet full of these foods will be harmful over the long term.
The are many substances present in food which claims to be fat free that aren't conducive to good health. I'm thinking specifically about sodium, and sugar, and high fructose corn syrup. There genetically modified organisms which are fat free. We don't know what the long term effects of eating genetically modified foods are, but we don know the effect of too much consumption of sugary and salty foods.
With the exception of plant foods that are naturally fat free, fat free foods are not particularly healthy. With processed foods with "fat free" on the label, the manufacturer has made up for the missing fat by adding sugar. Overeating sugar is just as bad for you and just as fattening. Also, fat-free food isn't as filling, making it easier to overeat. While overdoing it on fat isn't good for you, it is usually better to have a little fat, make the food more filling, and stop eating sooner.