Ovens-Covering the flame:[1]

There is no prohibition of Hatmana to leave food in an oven into Shabbos.[2] However, if the food is less than half cooked, then the heat source of the oven must be covered, due to the Shehiyah restrictions. Some Poskim[3] rule all the walls of the oven must be covered. Other Poskim[4], however, rule it suffices to cover one wall, and the knobs which turn the flame. Practically one should avoid using an oven for less than half cooked foods. According to all one may use a tin oven insert[5] which surrounds the food from all sides.

Shabbos oven: Those ovens which contain a Shabbos setting[6] must likewise be covered in order to leave less than half cooked food on them before Shabbos.[7] However there are Poskim[8] which say that if it is placed on this setting it does not need to be covered.[9]

Opening the oven on Shabbos: Regarding opening an oven on Shabbos if it does not have a Shabbos mode, see Chapter 7 Halacha 1!

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^[1] Shabbos Kehalacha page 312

[2] 2^nd opinion and final ruling in Admur 257:10; Rama 253:1; M”A 257:18; In Rishonim: Or Zarua 2:8; Ravaya 197; Mordechai 299

The reason: It was not forbidden to insulate on something which increases heat unless one attaches a moveable item around the pot, and the pot is thus completely insulated within that item. However, by an oven, even though the walls of the Tanur or Kira surround the walls of the pot, and [furthermore] even if the opening of the oven above has been closed, nevertheless since there is no moveable object which has been attached around the walls of the pot, this is not considered [the prohibition of] insulating, for the reason to be explained. Their reasoning is because the Sages only prohibited insulating with material that adds heat because of a decree that [if this were to be allowed then] one may come to insulate in embers. Therefore [insulating] is not prohibited unless it is done in a way similar [to insulating with] embers, [meaning that just like ember] is moveable and is attached to the walls of the pot [so too all insulation is only prohibited if it is moveable and attached to the walls of the pot]. [Thus, being that] the walls of the oven and its opening are not attached to the pot [therefore it is not prohibited to be done]. ([This applies] even if the oven is small and narrow, as it is not possible that there isn’t a small amount of air separating the [walls of the oven from the pot]) [Admur ibid]

[3] Chazon Ish 37:9, Minchas Yitzchak 3:28 and Igros Moshe 4:74-27

^[4] Sheivet Haleivi

[5] This is similar to a tin box with an opening.

[6] This places the oven on a set temperature.

[7] As one can easily switch the oven from a Shabbos setting to a non-Shabbos setting, just as we suspect one may come to raise the flame. What the Shabbos setting is useful for is to allow removing food from the oven on Shabbos without worry that doing so may lead to the fire turning on, or staying on longer. [Shabbos Kehalacha 8:17]

[8] SSH”K 1 footnote 17

[9] It does not suffice to claim that a Shabbos setting serves as a reminder that one does not raise the flame, as the Sages did not suffice with a mere reminder but rather with an act that diminishes the flame, as is evident from the concept of Ketumah. Perhaps however one can claim that placing it on a Shabbos setting diminishes the flame, and hence it can serve as a reminder.