Originally Posted by Mike Mills

No, it will not. However, I do not advise throwing it down the sink. Put it in a capped, glass jar and reuse it. The dirt will sink to the bottom and the oil will remain dissolved in the liquid. Decant the liquid for the next cleaning session. You can wipe the bottom of the jar out with a rag/towel and throw it away, taking the dirt with it.

Alternatively, let the mineral spirits evaporate. The amount used (and reused) for chain cleaning is trivial compared to what goes into the air from painting.