If I lose weight, what actually happens to the fat that I burn? Do I just sweat/burp/fart/pee/poop it away?
Your body will use the fat for energy, to maintain basic metabolic needs and for any exercise you care to do. Most of the calories (80%) in exercising produce heat, which is why you get warmer when you exercise. So you will sweat more and lose water and electrolytes that way. (Your body will produce more sweat to cool you off.)
The primary byproducts of fat metabolism is carbon dioxide (which you exhale) and water (which goes out the usual way.)
You don’t lose fat via sweat. Fat is turned into CO[sub]2[/sub] and a little water. You don’t have to sweat (or even exercise) to burn fat. It happens automatically if you have an energy deficit.
There’s an advertising trope in fitness right now that is probably confusing the multitudes: “Sweat is fat crying.”
OK, thanks!
I have a running shirt which says pretty much the same thing: “Sweat is just your fat cells crying”. Amusing, but I know it’s not literally true. 
Well, it is all water that needs to leave your body somehow. No doubt some of the water composing sweat came from fat burning. More of it will be stuff you drank, of course, plus a good bit from burning other nutrients. But yes, sweat is not a particularly important way of losing excess water. You probably lose more in the water vapor you breathe out, together with the CO2, not to mention peeing. Sweating as such is not necessary to lose weight, but it is probably a good indicator that you are exercising.
Most of the actual weight of the fat, though, will leave as CO2 as, by weight, fat is mostly carbon.
You breath in burned fat at a gym. Damn.
Just remember: if you can smell the shit of the guy in the next stall, it’s already inside you.
So if you can smell the bull…
What you smell is a few molecules of some volatile components of the fecal mass - a bit of indole, skatole, some thiols and hydrogen sulphide. At very low concentrations (we detect hydrogen sulphide at less than one part per billion). The actual shit (fibre, bacteria, undigested corn etc) is not volatile, unless aerosolised. You are not breathing it in when you smell it.
“I’m not fat, I’m just reducing my carbon emissions” 
I didn’t mean to imply that fat is converted into water. Water has no calories. Fat has 9 calories per gram. Fat is stored energy. If all the calories (in whatever form, sugar or protein) is not used immediately for the energy you need for metabolic processes, it is stored as fat. ATP (adenotriphosphate) is also stored energy, and is the immediate source of the energy you need. The breakdown of ATP into phosphate and other chemicals releases the energy needed to perform work. But energy is required to make ATP, and this energy is from the calories you either ingest or have stored. However, 80% of the calories used in work is lost as heat.
But as you pointed out, one of the products of fat metabolism is water. Fat consists of C, H and O. When combined with O[sub]2[/sub] from the air (“burnt”) it yields CO[sub]2[/sub] and H[sub]2[/sub]O, liberating energy.
You beat me to it!
All of the calories used eventually end up a heat, although some of it will enable you to do things such as moving your muscles along the way. The calorie is actually a measure of heat, and when the calories in food (or fat) are stated, what is being said is how much heat they will produce if burned.
The original question, though, did not seem to be about calories, but about where the matter, the weight (or rather, mass) of the body fat goes to when someone loses weight. The answer, as has been said, is that it is combined with oxygen and turned into CO2 (mostly) and water, and the CO2 is breathed out and the water (mostly) peed out.
When he said “calories lost as heat”, I don’t think barbitu8 was speaking in a broad physical or cosmological sense but rather in terms of the heat generated internally. I could believe that 80% is about the right number. The other calories you burn go to doing work on the environment.
Sure, it all ends up as heat on some time scale, but I guessing that’s not what you or he meant (eh?).