I don't know that we have a canon answer to this (If there were one, it would probably come from Sanderson's web site, and I didn't see anything on there.) We can make some logical deductions, though, based on what we know.
- Mistings can safely ingest certain types of elemental or alloy metals
- Mistings can perform some controlled bodily function, which they call "burning", that consumes the metals and releases various types of magical energies.
- The purity of the metals (either elemental purity or proper mixture of the alloys) is important.
- Impure or improperly mixed metals can be ineffective or even fatal.
- Mistings that hold "unburned" metals internally for too long risk losing them or getting metal poisoning.
- Once the metals have been "burned", they cannot be reused or recovered.
- This last bit is a bit of an assumption on my part, but given the scarcity of certain metals, I think if there was a "recycling" mechanism we'd have heard about it.
- Metals that have been "burned" are not fatal.
Given all of this, the most likely explanation for what happens when metals are "burned" is some form of metabolism, similar to but distinct from digestion. It must be distinct from digestion because simply swallowing and digesting the metals results in the same poisoning as it does for "normal" people. However, the behavior of "burning" the metals has all of the hallmarks of a chemical reaction: the metals change form, release energy, and impurities in the material affect the outcome. (They must change form somehow, or else they would remain poisonous.) This could be as simple as forming some inert compound as part of the chemical reaction.
Like any other by-products of metabolism, the most natural result of this process would be for the Misting's body to extract whatever energy it could from the metals and excrete whatever's left over as waste.
Given that it's magic, of course, anything is possible; the metal may be consumed in its entirety, or it could undergo something more like fission and turn into completely different elements. But the metabolic processing seems the simplest explanation.