Well, the good news is that it probably isn't going to do you any harm, and depending on the type of fat burner you may actually have some positive results, although they are going to be much less impressive. If you are mostly taking a fat burner that is focused on appetite suppression then you're obviously going to get better results than if you take one that's designed to help boost energy levels whilst working out and dieting. 

Can I Take Fat Burners Without Working Out And Lose Weight? 

So, as we mentioned above, yes you can, but if you're taking an energy boosting fat burner, then you're not going to get very much or any results. Most fat burners are designed to help you maintain energy levels when you're on a calorie restricted diet and excersising. You may find that you actually end up having trouble sleeping as a lot of fat burners contain caffeine, if you're not doing anything to burn off the extra energy. 

Then there's thermogenic fat burners, these should in theory raise your metabolism slightly, but this is usually the smallest effect that fat burners have. 

In short, all the studies show that best results from fat burners come when you combine them with diet and excersise as they're mostly designed to assist that. Not designed to work by themselves. And you should probably not trust anything that claims it can do all the work for you. So to answer the question of do fat burners work without excersise, the answer is not really, but they could help suppress appetite if you go for the right ones, but you'd do better by simply eating some low calorie high volume foods instead.