I forgot about the sodium iodide in acetone test.
In the case of the Bellstein test, one makes a copper halide - which burns a certain color (copper halides are used in fireworks).
CuO + 2 R-X (X is the halogen) -> CuX2 + R2O
In the silver nitrate test, silver abstracts halides... this salt tends to precipitate out of solution.
Ag+(aq) + X-(aq) -> AgX (s)
In the case of sodium iodide, it is a bit soluble in acetone, whereas sodium chloride isn't. So, R-Cl + NaI -> NaCl... so you'll see a white solid precipitate out of the solution if there is chloride in there.