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Is that true, though? Your body needs energy for various tasks and those have different mechanisms of spending the energy. Muscles, for example, move, which creates heat. But that heat is not simply breathed out.
Producing heat isn't where the mass goes though - mass is conserved. You only lose mass to energy in a nuclear reaction.
Something has to go in there, if not losing energy to radiant heat transfer, then how e=m(c^2)?
I'm not sure what you mean by in there but yes, the heat would be transferred to the environment.
E=m(c^2) describes how much energy is contained in matter. It's useful for nuclear reactions, but your body isn't a nuclear reactor and you aren't consuming substantial quantities of radioactive isotopes, like uranium ore, that will decay on their own so it isn't relevant here.
Still energy is being radiated. A mass loss has to occur for that
Radiation of heat is done through em waves which are massless particles. Being in direct contact with the air will transfer heat via conduction, or particles vibrating against each other - which is how the vast majority of heat loss will occur.