When you lose weight, how primarily does the excess mass leave your body?
I've been looking online for an answer to this question, and the search itself has been interesting.
For one thing, a lot of people obviously don't understand the principle of conservation of mass, or if they do, they have a blind spot about applying it to the human body. It turns out that until some pioneering medical research was done in the 1800s by Wilbur Olin Atwater, even many scientists were inclined to think of the human body as something special and different to which the known laws of physics might not necessarily apply.
In discussion forums where this topic came up many people were happy to simply say that your excess weight is "burned away" or "turned into energy", as if saying that was enough to answer the question of how excess mass leaves your body. Once something is "burned" or "turned into energy", of course, it's mass doesn't just magically disappear.
(While it's true that mass can be converted into energy via the famous E = mc², for bodily chemical reactions the effect is insignificant -- unless perhaps you've had your intestines replaced by a small, portable nuclear reactor.)
The answer to my question apparently lies in a combination of biological processes like the citric acid cycle, AKA the Krebs cycle, which, along with a few other steps, means that the proteins, fats, and carbohydrates you consume are, one way or another, reduced to glucose, and then (at least in the aerobic part of your metabolism, which is the largest part of your metabolism) oxygen is combined with glucose, and carbon dioxide and water are produced.
Since the main byproducts of our metabolism are carbon dioxide and water, it would then follow that exhalation and urination and perhaps perspiration would be the primary ways stored fat leaves your body when consumed. When you consider that the oxygen component of the C0₂ and H₂0 isn't stored in your body fat, but obtained from the air as needed, and that hydrogen weighs much less than the carbon, the greatest mass is going to leave your body as C0₂, with the C part of that being what used to be your body fat.
It would follow that when you lose weight, you primarily exhale the weight away. It's odd to think of breathing away pounds!
I'd love to find a more thorough and more detailed accounting for how what goes into our bodies eventually comes out of our bodies, something that takes into account other waste products, the contributions of anaerobic metabolism, etc.
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...you're exhaling them away. 
Generating body heat and bodily motion can, of course, help you lose weight, but the weight doesn't just magically turn into or disappear into "heat and motion".
Burning food and stored body fat for energy, although it involves a very sophisticated chemical dance, in the end turns out to be very much like burning a log to produce heat. A burned log doesn't just vanish into "heat", it mostly turns into carbon dioxide and water vapor. If you carefully collect all of the ashes, all of gasses given off by the fire, all of the soot thrown into the air, and weigh it, it will all weigh just as much as the original log, save a probably immeasurably small loss of mass in accord with E=mc² (probably well under a picogram, to take a wild guess, definitely far, far less than even a single gram).
Loss of mass from the production of energy does not become significant until nuclear reactions are involved.
I said in my OP, "For one thing, a lot of people obviously don't understand the principle of conservation of mass, or if they do, they have a blind spot about applying it to the human body." Not to pick on you, but your guess appears to make you one of those people.