Halacha forbids placing any kind of food on an open flame on Shabbat. Even though the food has been fully cooked, and one simply wants to reheat it, he may not place the food on an open flame. Thus, for example, it is forbidden to place a pot of rice over an open flame on the stove on Shabbat to reheat it, even though it had been cooked before Shabbat.
Warming food on Shabbat is permissible only if one uses a "Blech," a metal sheet covering the stove, or a hotplate, which has a metal surface covering the heating element. Once the fire is covered, and not exposed, it is permissible to place certain foods on the covering on Shabbat. Namely, one may, according to our custom, place solid food that has been fully cooked on a Blech or hotplate on Shabbat. For example, it is permissible on Shabbat morning to place a pot of rice on a Blech or hotplate to be reheated. Even though the pot has been in the refrigerator all night and the food is cold, one may place the pot on the Blech or hotplate, since the fire is covered.
One may not, however, place cold liquid on a Blech or hotplate on Shabbat, even though it has been fully cooked before Shabbat. Halacha follows the view that "Yesh Bishul Ahar Bishul Be’lah," which means that once a liquid has been cooled, reheating it constitutes "Bishul" ("cooking") with respect to the Shabbat laws. As such, it is forbidden to reheat liquid on Shabbat regardless of whether the flame is exposed or covered. Thus, for example, one may not place cold Kibe Hamda (soup with meatballs) on a Blech or hotplate on Shabbat. Similarly, if one has roast in gravy, he must pour out the gravy before warming up the roast on the Blech or hotplate.
One who has cold liquid food, such as Kibe Hamda, which he wishes to eat hot on Shabbat, may ask a gentile – such as a non-Jewish housekeeper – to place it on a Blech or hotplate. Generally, Halacha forbids asking or instructing a gentile to perform on Shabbat an action which is forbidden for Jews to perform. However, in the case of reheating cold liquid food on a covered flame, the Hida (Rav Haim Yosef David Azulai, 1724-1806) rules that it is permissible to ask a gentile. The reason for his ruling is that issue of reheating cold liquids on a covered flame on Shabbat is subject to a debate among the Halachic authorities. Although we follow the stringent position, which forbids reheating such foods, we may rely on the lenient view with respect to "Amira Le’nochri" – asking a gentile. Therefore, despite the fact that one may not personally reheat cold liquid on a Blech or hotplate on Shabbat, it is permissible to ask one’s non-Jewish housekeeper to do so. This ruling is cited as Halacha by Hacham Yishak Yosef in his She’erit Yosef (vol. 3, p. 405; listen to audio recording for precise citation). See also Yabia Omer, vol. 7, p. 131.
It must be emphasized, however, that this Halacha is limited to the particular issue of reheating cold liquids. One should not extrapolate from this ruling that one may ask his housekeeper to also perform other forms of cooking and reheating. As discussed, the case of reheating cold liquids is exceptional, due to the fact that some Medieval authorities deemed it permissible even for Jews.
Summary: One may not place any food – even fully cooked food – on an open flame on Shabbat. One may, however, place solid food on a Blech or hotplate on Shabbat, provided that it had been cooked before Shabbat. One may also ask a gentile to reheat liquid food (that had already been cooked) on a Blech or hotplate on Shabbat.