I discussed this question of what happens to water conditioner over time with Neale Monks, a name that is (or should be) familiar to everyone in this hobby. Dr. Monks has degrees in biology and paleontology, has written probably hundreds of articles, and as the adage goes, whatever he doesn't know about the hobby is not worth knowing. The following is a summation of our discussion.
Conditioners neutralize chlorine/chloramine and heavy metals, and in both cases what it is doing is binding, hopefully irreversibly under normal conditions, to those chemicals. What is created is a new, soluble compound (or set of compounds) that stay dissolved in the water. Being non-toxic, they effectively become added to the total dissolved solids (TDS), but in such trivial quantities we don't have to worry about them. Sodium thiosulphate is the default dechlorinator, and reacts with chlorine to form sulphur, hydrogen chloride, and a salt called sodium sulphate. The sulphur isn't soluble and presumably precipitates out or else reacts with other chemicals in the water. But HCl and NaSO4 are very soluble, the first very acidic the second being neutral, but since both are in tiny, tiny amounts their impact on water chemistry will be nil.
I mentioned that Seachem have told me the effectiveness of Prime lasts 24-36 hours with respect to the heavy metals. Neale surmised that likely the EDTA used in Seachem Prime either breaks down or is metabolised by filter bacteria, so either way, can't be guaranteed to maintain the necessary concentration beyond 36 hours. But I don't know for sure.
As for what happens to the "conditioner," Neale said "nothing; it's just floating about, as new compounds or at least complexes, in the water. It does not likely dissipate out, or out-gas in some way. It would be removed as TDS at the next water change.
Substances dissolved in the water, like the conditioner, get inside the fish via osmosis if the substances are small enough particles to diffuse through the cell membranes. The fish's physiology will deal with it over time. Freshwater fish will be excreting a great deal of water because their cells are hypertonic to their environment (i.e., the water they are swimming in). Unwanted water-soluble chemicals will be excreted along with that water. While their kidneys are differently arranged to ours in some ways, their basic function and many of the tissues are essentially the same.