Why does molten salt turn black?
Part of the color change might be due to creation of F-centers, crystal defects where there are electrons filling the place of an atom. A corresponding H-center (for halogen interstitial) may also be created for each F-center.

That said, such color changes tend to be rather pale, and as you state, $\ce{NaCl}$ heated to melting in an iron crucible may appear dark or even black. This is likely due to dissolution and oxidation of the crucible. The molten salt attacks the metal at grain boundaries and also dissolves protective oxide coatings, allowing oxygen from the air to attack the underlying metal. Even in a closed system, such as a molten (mixed) salt thermal energy storage system, corrosion is a problem.

The color change could also be due to trace impurities in the "table salt". For example, sodium or potassium ferrocyanide, $\ce{Na4[Fe(CN)6]·10H2O}$, is intentionally added to keep the salt free-flowing despite humidity changes. This might react at high temperatures to form a deeply-colored compound.

To find out what is causing the discoloration, you might try heating a small sample of high purity $\ce{NaCl}$ in a ceramic or platinum (a bit pricey) crucible, and you could analyze heated table salt for dissolved iron compounds.

BTW, F-centers and other crystal defects can be created intentionally with optical lasers and x-rays. If memory serves, there was an article in Scientific American's Amateur Scientist about creating F-centers using an electrostatic generator.