Mass is simply a form of energy, converted as E=mc^2. Energy is what's conserved. The model breaks down dramatically once you're doing dramatic forms of mass-energy conversion, like with nuclear reactions.
EDIT: I appreciate the correction on my misinformation about other types of reactions, and will not continue to spread that untruth here. Thank you, Locke1689.
"Even simple exothermic chemical reactions convert mass to energy by the same ratio as nuclear reactions."
That's not exactly true. Most exothermic reactions are simply the restructuring of chemical bonds (molecular bonds) to lower energy states. Consider that every molecular bond stores a certain amount of chemical potential energy. By breaking that bond, the potential energy is released in the form of kinetic energy, or heat.
There is no fission or fusion in normal chemical exothermic reactions and no matter is created or destroyed. Instead, the process is more similar to the powering of a motor via electricity -- electric potential energy is converted into kinetic energy through the actions of electromagnetism.
EDIT: I appreciate the correction on my misinformation about other types of reactions, and will not continue to spread that untruth here. Thank you, Locke1689.