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1. Insulating food on Shabbos itself:
The prohibition: On Shabbos itself it is forbidden to insulate hot foods even with material that does not add heat.
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. [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.]
May one insulate on Shabbos food which further cooking will cause it damage: 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.
If the pot was in the oven before Shabbos may it be insulated on Shabbos? If the pot was in the Kirah and on Shabbos one comes to take it out from the Kirah 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 257:2
 Admur ibid; Michaber 257:1
 Admur ibid; Shabbos 51a; Beis Yosef in name of Rashi; M”A 257:1 in name of Rosh; M”B 257:1
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.
 Shaar Hatziyon 257:3
 Admur 257:3; Michaber 257:2
 Admur 253:14