May one remove the cover from a pot food that is on the fire on Shabbos and then place it back on?
Removing: One may always remove the cover of a pot of food on Shabbos even if the pot is sitting over a fire, or Blech, or electric plate. This applies even if the food is not fully cooked. The question however is asked regarding whether it is permitted to then replace the cover back onto the pot.
Replacing the cover onto the pot: If the food in the pot is fully cooked then one may replace the cover onto the pot even while it is sitting in the fire, and so is the custom. However, if the pot is not currently boiling and placing back the cover will cause it to re-boil, then there is room to be stringent, although one who is lenient certainly has upon whom to rely. If, however, the food is not fully cooked placing the cover on the pot is forbidden according to all.
Making sure that the inside of the cover is dry, or still warm, by them time is replaced: Even when the food is fully cooked and one is thus allowed to return the cover onto the pot while it is on the fire, one must beware not to return the cover to the pot if the cover is wet [which is very common due to vapor] unless the moisture of the cover is still hot, or one dries the cover. [A simple solution to keep the cover from cooling off is to sit it on the Blech.]
 See Shabbos Kehalacha Vol. 1 p. 230-232; Piskeiy Teshuvos 318:56, 58; Igros Moshe 4:74-10 rules that this question depends on the reason behind Meigis, and that according to the reason that it is due to decree that one may come to do so even by uncooked food, then here too it should be forbidden. However, according to the other reasons, it would be allowed.
 Setimas Kol Haposkim Baalei Hashulchan Aruch who all omit any such prohibition despite it being very common; Ketzos Hashulchan 124 footnote 10; Igros Moshe ibid according to Poskim who rule the reason of Meigis has nothing to do with a decree of “Atu Eino Mevushal”; Shabbos Kehalacha 5:16; Mara Deshmaasah 122; Maor Hashabbos 2 Letters of Rav SZ”A 19:1 and Rav Fisher and Rav Elyashiv; Az Nidbaru 1:79; 5:13; Teshuvos Vehanhagos 1:297; Piskeiy Teshuvos 318:58; See Admur 254:4 and 257:8 that one may return fallen insulation onto the pot if the food was fully cooked
Other opinions: Some Poskim rule that it is forbidden to cover a pot of food while it is on the fire even if it is fully cooked, due to the Meigis prohibition and the fact that it appears as if one is cooking, and may come to do so to an non-fully cooked food. [Makor Chaim of Chavos Yair 318; Tehila Ledavid Tinyana 252; Shevisas Shabbos Mevashel 81; Igros Moshe 4:74-10 according to Poskim who rule the reason of Meigis is because of a decree of “Atu Eino Mevushal”; Piskeiy Teshuvos ibid footnote 531] Some Poskim conclude based on this that it is proper to be stringent. [Vayaan Yosef O.C. 159; Igros Moshe ibid; SSH”K 1 footnote 108]
 Ketzos Hashulchan ibid; Rav Farkash in Shabbos Kehalacha 5:16 writes that one may be lenient even in such a case
The reason: The Shevisas Shabbos ibid rules that covering even a fully cooked food that is on the fire is forbidden due to “Meigis” mixing. The Ketzos Hashulchan ibid however limits this ruling only to a case that covering the pot will cause it to re-boil, as only in such a case is there mixing taking place. Practically, the Ketzos Hashulchan leaves this matter in question [even in a case that covering will cause it to re-boil], although leans to be lenient.
 Admur 254:4 and 257:8; Biur Halacha 257:4 “Goreim”; Ketzos Hashulchan 124 footnote 10; Piskeiy Teshuvos ibid and ibid
The reason: Doing so is forbidden due to the cooking prohibition, being that it hastens the cooking.
 Shabbos Kehalacha Vol. 1 p. 285; Piskeiy Teshuvos ibid footnote 533
 Even if some of the moisture accidentally spills onto the Blech and causes it to sizzle, we do not find any recorded prohibition in doing so, as the moisture is already fully cooked and does not contain a looking like you’re cooking prohibition.