1. The prohibition against insulating food on Shabbos:
It is forbidden to insulate hot foods on Shabbos. This applies even if the insulation material does not add heat. This applies even if the food is fully cooked and even if further cooking will cause it damage. This applies even if the food was left on the fire from before Shabbos in a permitted way. [Thus, it is forbidden to insulate a hot pot of food on Shabbos even if the pot is on one’s counter and not on a heat source.]
Materials that add heat: It is forbidden to insulate using material which adds heat, even completely cold food, and even before Shabbos, as explained in the previous chapter.
If the insulating material does not directly touch the walls of the pot, then it is not considered insulated. See Chapter 3 Halacha 3B for the full details of this matter.
B. How much of the pot must be insulated for it to be considered a prohibited insulation?
If the top of the pot is covered with insulation material [whether the material adds heat or does not add heat] then it is defined as “insulation that does not add heat,” and may thus not be done on Shabbos even if the sides of the pot are not covered with the material. Accordingly, it is forbidden to place a towel or the like on top of a hot pot on Shabbos, whether it is on or off the fire, as doing so is defined as insulation. However, if the pot is off the fire, then it is permitted to surround completely its walls with a towel, leaving its top [meaning its cover] uncovered by the insulation material. However, if the pot is sitting on top of a source of heat [such as a blech, or on top of a pot that is on the blech] then in addition to it being prohibited to place a towel over its top, as well one may not wrap a towel around majority of the pots walls, as this is considered insulation which adds heat which is forbidden to be done even before Shabbos! As well if the food is not fully cooked one may not place insulation even on minority of its walls.
 Admur 257:2; Michaber 257:1
 The reason: [Now,] even though [insulating with material that does not increase heat] is not similar to [insulating with] ember at all, and thus why then is it forbidden? [Nevertheless, it is forbidden] due to a decree that perhaps when one comes to remove the pot from the stove in order to insulate it with material that does not increase heat, he will find that the pot is cold, and will forget that it is Shabbos and reheat it on the fire, hence being liable for [the prohibition of] cooking. [Now this suspicion only applies according to those which hold that reheating a cooked food constitutes cooking, however] according to those which say that there is no cooking [prohibition involved] with [reheating] something that was already cooked even if it became cold afterwards, as will be explained in chapter 318, [nevertheless even according to them insulating on Shabbos is forbidden due to a different worry as] we are worried that one may stoke the coals in order reheat it and will thus be liable for [the prohibition] of making a fire. [Admur ibid; Shabbos 51a; Beis Yosef in name of Rashi; M”A 257:1 in name of Rosh; M”B 257:1] Thus, it is forbidden to insulate even foods that are dry and do not have a cooking prohibition, due to the above worry of stoking the coals. [Shaar Hatziyon 257:3]
Other opinions: Some Poskim rule that the reason for why insulating on Shabbos is forbidden is because one may come to insulate in ember. [Rambam Hilchos Shabbos 4:3] See Lechem Mishneh ibid that explains the source of the Rambam from the above Gemara and how he interprets it differently.
 Admur 257:3; “Even food which is completely cooked, and further cooking will condense it in a way that is damaging to it, is forbidden to be insulated from when it becomes dark [after Bein Hashmashos, by the definitive beginning of Shabbos], even with a material that does not add heat”; Michaber 257:2
 Admur 253:14 “If the pot was in the Kira and on Shabbos one comes to take it out from the Kira and insulate it, even if he wants to use a [form of insulation, such as] pillows or blankets or other clothing, which does not add heat but just retains the heat, it is [nevertheless] forbidden, as it is considered like one is insulating the pot now on Shabbos for its first time, which is forbidden [to be done] even if the insulation does not add heat.”
 Admur in Mahadurah Basra 259; Ketzos Hashulchan 71 footnote 34; Shabbos Kehalacha Vol. 2 page 56-58; Chazon Ish 37:19 forbids any form of insulation, even minute, on Shabbos; See Piskeiy Teshuvos 257:12
Background: This ruling is based on the explanation of the Alter Rebbe in the Mahadurah Basra that there is a difference between insulation that adds heat and insulation that does not add heat, that by the latter, the main aspect is the top of the pot, as that is how the heat is mainly guarded. This ruling in the Mahadurah Basra thus seems to be a retraction of his ruling in 257:4, and in the summary brought in chapter 259. This ruling is brought as well by the Ketzos Hashulchan chapter 71 note 34. Other Poskim however rule that by this form of insulation it is only problematic if the entire pot has been covered. So rules also the Shemiras Shabbos Kehilchasa, following the Alter Rebbes ruling in his Shulchan Aruch.
Ruling of Admur in Shulchan Aruch: It is only forbidden to completely insulate the pot within the insulation material on Shabbos. [Admur 257:4] Even according to the Michaber we do not rule Hatmana Bemiktzas regarding insulating on Shabbos. [Kuntrus Achron 257:2]
Other opinions: Some Poskim rule that this follows the same law as Hatamana on Erev Shabbos and hence according to the Michaber it is forbidden even to partially insulate the pot while according to the Rama it is only forbidden if completely insulated. [Beir Heiytiv 253:17; M”B 253:69] Others rule that according to all it is only considered insulation if it is completely insulated. [Kuntrus Achron 257:2; SSH”K 1:66; Or Letziyon 17:12] Practically the custom of the world is to be lenient like the Rama. [Piskeiy Teshuvos ibid]
 This latter ruling holds true even according to the Alter Rebbes SH”A as stated in 253:14: “However if [on Shabbos one wants to insulate the pot by having] the pillows and blankets wrapped [only] around the walls of the pot, but leaving its top revealed, then it is permitted to be done as this is not considered [the prohibited form of] insulation.”