The Shehiyah[1] restrictions and its reason:^[2]

In certain cases, the sages decreed against leaving uncooked food over a flame into Shabbos because if one were allowed to leave the food on the fire once Shabbos began, one may forget that it is Shabbos and stir the coals [i.e. increase the flame[3]] on Shabbos in order to finish the cooking.^[4] Now, in which cases did the Sages make this restriction? The Sages only suspected that one would come to raise the flame under his food, if the food was under cooked to a certain point, and if the fire being used was of a type that there is room to suspect that one may come to instinctively raise the flame. [In other words, the prohibition of the sages against leaving food over a fire into Shabbos only applies by one who has a) a problematic food and b) a problematic fire. However, if the food is problematic but not the fire, or if the oven is problematic, but not the food, then no restrictions were placed.] Now, what food and what fire is defined as problematic? In a nutshell only a food that is less than half cooked is defined as a problematic food and only an uncovered flame is defined as a problematic fire. So, if the flame is covered or the food is half cooked then the food may be left there from before Shabbos, without any restrictions. These details will be elaborated on in Halacha 3-4

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[1] Shehiyah literally means to leave and refers to leaving a food on a fire over Shabbos, having been placed there on Erev Shabbos for the Shabbos meals.

^[2] Admur 253:1; Michaber 253:1

[3] Does the suspicion of stoking the coals apply by today’s ovens and stoves: There is a discussion amongst Poskim regarding if in truth the suspicion of “Shema Yechta” applies today by our ovens and stoves being that one can only higher the flame, while the suspicion back then was not that one may come to add more wood to the flame but that he may blow on the coals. Thus, some Poskim side that today by our gas and electric cooking ovens and stoves the suspicion is no longer applicable. [Igros Moshe 1:93] Nevertheless even the Igros Moshe concludes that the flame must be covered, and so rule all the Poskim today. [Shabbos Kehalacha Vol. 1 page 296 Biurim 4]

[4] The stoking of coals was very common in prior times and hence the Sages suspected one may momentarily forget that it is Shabbos and come to stoke the coals out of instinct. [Aruch Hashulchan 253:1]

What does one transgress by stoking the coals? One transgresses the prohibition of Maavir. [257:2] Alternatively, if the food is not yet cooked one will transgress the cooking prohibition. [Aruch Hashulchan ibid]