A question recently arose concerning the situation of a guest who was served hot soup during Shabbat lunch. The guest was surprised to see hot soup being served on Shabbat day, as Halacha forbids reheating cold liquid on a Blech or hotplate on Shabbat, a law known as “Yesh Bishul Ahar Bishul Be’lah.” It turned out that the hostess mistakenly thought that one may reheat hot soup on Shabbat, and thus accidentally violated this Halacha. The question thus arose as to whether the soup was permissible to be eaten. Generally speaking, Halacha forbids deriving benefit from prohibited acts committed on Shabbat. Seemingly, then, it would be forbidden to eat the hot soup, as one would thereby be deriving benefit from a Shabbat violation.
However, Hacham Ovadia Yosef, in his Hazon Ovadia (vol. 4, p. 422; listen to audio recording for precise citation), ruled that the soup may be eaten in this case. He writes that although food that has been prepared in prohibited fashion on Shabbat is forbidden for consumption, this applies only when the act in question is forbidden according to all authorities. When, however, the permissibility of the act is subject to a debate among the Halachic authorities, then the food is not forbidden. Hacham Ovadia writes that in such a case we may apply the rule of “Safek De’rabbanan Le’kula” (we may act leniently in situations of doubt concerning a Rabbinic prohibition). Since deriving benefit from food cooked on Shabbat is forbidden only by force of Rabbinic enactment, the food is permissible if there is some question concerning the act that was performed. This important principle applies in the case of liquid that was reheated on Shabbat. Although the Shulhan Aruch rules that it is forbidden to reheat liquid on Shabbat, and this is the accepted Halacha, there were some Rishonim (Medieval Halachic scholars) who held that once a liquid had been cooked, it may be reheated on Shabbat even after it had completely cooled. As mentioned, we do not follow this position, but nevertheless, since this issue is subject to debate, if soup was reheated on Shabbat it does not become forbidden.
Summary: It is forbidden to reheat cold soup on Shabbat, but if one mistakenly did reheat soup, it may be eaten.