One of the thirty nine Melachot-prohibited activities on Shabbat is "HaOfeh"-baking. Although baking was not actually done as part of the construction of the Mishkan, cooking was done. They would cook the various plants to extract the dyes for the fabrics and skins. The Hachamim explain that cooking and baking are the same process-using fire to change the consistency of an entity, by making hard objects soft or soft objects hard. Therefore, both cooking and baking are prohibited by the Torah.
The Mishna in Masechet Shabbat (p. 38) states that one may not place an egg next to a hot urn in order to roast it. That is, not only is cooking directly from a fire prohibited, but even cooking via surfaces which were heated by a fire, in this case the metal of the urn. The Talmud Bavli understands this to be a Torah prohibition. The Rambam in Hilchot Shabbat (Ch. 9) and Shulhan Aruch (318:3) rule in accordance with this Gemara that there is no difference between cooking with "Ur"-fire and "Toldot HaUr"-indirectly cooking with thing materials heated by the fire.
It should be pointed out that the Talmud Yerushalmi argues with this assumption and holds that "Toldot HaUr" is only prohibited MDrabanan. The Halacha is in accordance with the Talmud Bavli.
SUMMARY
It is prohibited by Torah law on Shabbat to cook on surfaces heated by a fire.
