Strictly speaking, you would weigh more when hot than when cold. Relativity really does work that way. It’s just that the amount of difference would be so minuscule that there’d be no hope of actually measuring it.
See, I knew that increased CO2 levels, and therefore climate change, was man-made! 
All forms of energy have mass. A compressed spring has more mass than a relaxed one. A block sitting on a table has more mass than one sitting on the floor. A hot rock has more mass than a cold one. The technology to measure such tiny differences does not exist for these macroscopic examples. But potential energy-mass (from the strong force) is measurable in atomic nuclei, which have masses less than the sum of the masses of the same number of free protons and neutrons.
In other words, it disappears into thin air.
E=mc2 does not mean that energy has mass, it means that, under some circumstances, energy can be converted into mass (e.g., when a gamma ray photon produces an electron-positron pair) and vice versa (e.g., in a nuclear reactor). There may be a vanishingly tiny amount of relativistic mass lost when the chemical bonds of the fat molecules are broken, but this is utterly negligible compared to the weight of the atoms involved. Any measurable weight loss achieved by exercise come from the carbon, hydrogen and oxygen in the fat being excreted as CO2 and water.
I’m confused. When I tried to look this up, I got stuff about fatty acids turning into keytone bodies in a reaction that was more efficient than using glucose. I tried to figure out what keytone bodies turn into (since the brain can use that as fuel), and I could find nothing. Is that the point where it is split into nitrogen, carbon dioxide, and water?
Ketone bodies are converted into acetyl-CoA. Acetyl-CoA enters the TCA cycle and is converted into CO2.