Reheating food on Shabbos on top of a heat source:

Heating foods on Shabbos is only permitted if it does not transgress the Biblical cooking prohibition, as well as the Rabbinical heating prohibition. Certain precooked foods contain a Biblical cooking prohibition against reheating them, as we will explain in A. Furthermore, even foods that do not contain any cooking prohibition, the Sages regulated the methods in which one is allowed to heat it in, as will be explained in B. Accordingly, one may only warm a food on Shabbos if it is a type of food that is negated of a cooking prohibition, and if it is heated in a method that is permitted by the Sages. Both aspects will be discussed below:

A. The foods that may be reheated:

All foods that fulfill all the following conditions do not contain a cooking prohibition upon reheating them:

1. Fully cooked:[1] The food must be fully cooked, [to the point that it requires no further softening, or change of texture, through cooking]. If the food is not fully cooked, even if it is half cooked, it contains Biblical cooking prohibition against cooking it any further. It is hence forbidden to heat up such food on Shabbos.
2. Dry: ^[2] The food must be completely[3] dry and not contain any moisture at all. This means that a solid food must be dry to the point that it is not Tofeiach Al Menas Lehatfiach[4], which means that it’s moisture cannot wet one’s finger enough to make a further finger wet.[5] If the food contains cold moisture past this point, and certainly if the food contains actual cold liquid, it is forbidden to heat it on Shabbos due to the cooking prohibition, even if the food was fully cooked before Shabbos.
3. Does not melt:[6] The food may not contain congealed solids which will melt with the heat. If the food contains congealed solids that will melt with the heat, it is forbidden to be heated on Shabbos even if the food was previously cooked or roasted.

B. The permitted method of reheating:

All foods that do not contain a cooking prohibition upon reheating, as explained in A, nevertheless, retain a Rabbinical restriction against being heated in a method that appears like one is cooking the food on Shabbos. Accordingly, it is forbidden to heat the food by placing it directly over a flame.[7] Furthermore, even if one only momentarily removed the food from the flame on Shabbos, it may not be returned unless six conditions are fulfilled, as will be explained in another Halacha.[8] Furthermore, even if a Blech covers the flame, or one uses an electric plate, the consensus amongst many Poskim[9] is that it is forbidden to heat the food directly on the blech or electric plate. The only form of heating that remains valid, is to place the food on top of a pot that is on the flame.[10] The reason this method is valid is because it does not appear to anyone that one is trying to cook food in such a method.[11] One may either place the food on top of another pot of food that is already on the flame, or place an upside pot on top of the Blech or electric plate, and then heat the food on top of the empty pot.[12] The upside-down pot may be placed even on Shabbos.[13] However, it does not suffice to place an upside-down pot over an uncovered flame [i.e. no Blech] and then place the food on top of it to warm.[14]

C. Foods that were cooked in liquids:[15]

Any food which has fulfilled the above three conditions in A [i.e. fully cooked, dry and does not melt] does not contain a cooking prohibition through reheating it on top of a pot and the like, even if the food was previously cooked in liquid [i.e. a dry Matzah ball] and one now desires to reheat it without liquid. It is only forbidden to reheat a previously cooked food when one does so food directly over a fire, without a pot and the like which intervenes between it and the flame. [i.e. barbecue].[16]

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[1] Admur 253:18; 318:10; Rama 253:2; Michaber 318:4; M”B 253 Biur Halacha “Vedavka Shehatvshil”; Biur Halcha 253:2 “Mutar Lehachziro”

Other Opinions: Some Rishonim rule that further cooking of a food that has reached Ben Drusaiy is permitted on Shabbos. [See Biur Halacha ibid and 318:4; Even Yisrael 8:28; Piskeiy Teshuvos 253:18]

[2] 2^nd opinion and final ruling and custom in Admur 318:9; Michaber 318:4; Rama 318:15; Rashi; Rabbeinu Yonah; Rosh

Other opinions: Some Poskim rule that even if [the precooked liquid] has [already] completely cooled down there is no [prohibition in] cooking it after [it having been previously] cooked. [2^nd opinion in Admur ibid; Rama 318:15 ; Rashba; Ran; Magid Mishneh] The custom is to be stringent like the first opinion. [Admur ibid; Rama 318:15]

[3] 253:19; 318:11; 318:17; Chasam Sofer 74; Divrei Nechemia p. 36 in Hosafos; Ketzos Hashulchan 124 footnote 37; Igros Moshe 4:74:7; Or Letziyon 2:30:13

Background:

253:19: [Furthermore] even if the food [which one wants to return] is completely cooked but it has already completely cooled down, then it is forbidden to return it in all cases, even if it is a type of food which reheating it does not consist of a  prohibition of cooking, such as is the case with a dry food that has no liquid at all that is fully cooked,

318:11: Dry food which contains no liquid at all is [permitted to be] cooked after having been previously cooked as long as it was already fully cooked, even if it completely cooled down.

318:17: Nevertheless, ox meat is forbidden to place inside a Keli Rishon [that is off the fire] as although the meat [itself] will not cook, nevertheless the moisture that is on it will cook. Although if [the meat] is completely dry and does not have on it any moisture at all [then it is allowed].

Chasam Sofer 74: “Only if the coffee is completely dry and contains no moisture at all”

Divrei Nechemia p. 36: “This opinion [if the Minchas Kohen] is Muktzah Min Hadaas and is based on a mistaken Nussach in the Rabbeinu Yerucham.

Other opinions: Some Poskim rule that so long as majority of the food is dry [solid] it is permitted to place it on the fire even if it containes a minority of liquid. [Nussach of Rabbeinu Yerucham in Beis Yosef 253; Minchas Kohen Mishmeres Hashabbos 2:2; Iglei Tal Ofeh 34; P”M 253 M”Z 13; Kaf Hachaim 318:62] This is because when majority of the food is a solid one does not have in mind to cook the liquid and in such a case one may even initially rely on those Poskim that are lenient to say there is no Bishul after Bishul by liquids. [Iglei Tal ibid]

[4] Ketzos Hashulchan 124 footnote 37

[5] 613:14

[6] Admur in Siddur; Levush 318:16; Taz 318:20; See Shabbos Kehalacha p. 148-151

Other opinions: Some Poskim rule there is no Bishul after Bishul by a precooked dissolvable item. [Admur 318:11; 318:25; 318:31; Michaber 318:16; M”A 318:40]

[7] Admur 253:14; Michaber 253:2

[8] Admur 253:14-18

[9] Shabbos Kehalacha Vol. 1 page 411-414; Piskeiy Teshuvos 253:28-29 and footnote 264; SSH”K 1: ; Igros Moshe 1:93; 4:74-32

Other Poskim: Some Poskim are lenient to permit heating all foods on a Blech. [Zera Emes; Mahram Brisk 2:76; Maharam Shick 117; See Shabbos Kehalacha 10 footnote 23 p. 412; See Piskeiy Teshuvos 253 footnote 264 in name of Degul Merivava]

Areas on the blech that are not directly over the fire: Have the same status as areas which are directly over the fire.

[10] Admur 253:26; Michaber 253:3

[11] Admur ibid

[12] See Rav Shlomo Zalman Aurbach SSH”K Tikkunim 1 footnote 112; Sheivet Haleivi 1:91; Shabbos Kehalacha Vol. 1 page 405-408; Piskeiy Teshuvos 253:25

[13] Admur ibid

[14] Michaber 253:3; Biur Halacha 253 “Veyizaher”; Admur 253:26 Kuntrus Achron 10; Piskeiy Teshuvos ibid

[15] Admur 318:15-16 [This refers even to a food that was cooked in liquids as is understood from the wording there and from the contrast to the wording in 318:24]; Tehila Ledavid 318:24; Chazon Ish 32:14; Az Nidbaru 8:10; Iglei Tal Ofeh 9; Shabbos Kehalacha 3:24 Biurim 18; Piskeiy Teshuvos 318:14

Other Opinions: The M”B 318:41 rules it is forbidden to place a cooked dry food on top of a pot and the like due to the prohibition of roasting after cooking. [See Piskeiy Teshuvos 318:14]

[16] Admur 318:13; 318:24; M”A 318:17; M”B 318:41