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A. Placing food next to the wall of a Kirah oven on Shabbos:
All the above restrictions are with regards to returning the pot onto the oven or inside it, however it is permitted to even initially place the pot next to the [Kirah] oven, on the outside of it, even if the actual food is touching the walls of the oven from the side, and even if it was fueled with olive waste and wood and it’s coals were not covered or swept as we are not worried that one may come to stoke the coals. However, this allowance is only as long as there is no prohibition of cooking involved here at all, such as that the food is fully cooked and dry [and does not contain fat], or is a fully cooked liquid which is still warm.
The reason that this is permitted is: because this does not appear like one is initially cooking the food on Shabbos, as it is not usual to cook a food outside of the Tanur or Kirah and it thus has the same laws of placing a food next to a bonfire as will be explained in 318.
Other Opinions: However there are those which prohibit to place a pot near a Kirah oven if their coals are not swept or covered and have been fueled with olive waste or wood, even if [at the time one removed it] one had in mind to return it or to place it next to the oven, and [even if] it is still in his hand.
Their reasoning to prohibit this is: because of a decree that [if one were allowed to do so then] he may come to stoke the coals, as even though with regards to placing food near an oven before Shabbos we are not worried that one may come to stoke the coals, nevertheless concerning placing it there on Shabbos they were stricter.
The Final Ruling: The custom is like the first opinion, however in a situation that there is not really much need for one to do so it is proper to suspect for the latter opinion [and not place food near a Kirah which has been fueled with olive waste or wood, if it has not been swept or covered]. [See Q&A 24-25]
B. Placing food next to a stove top/open fire?
Cooked foods and liquids: Any [food or liquid] which does not have a cooking prohibition [applicable to it], is permitted to even be boiled near a bonfire. For example a liquidly food which was completely cooked and is still hot to the point of Yad Soledes, or even if it is not [hot anymore to the point of] Yad Soledes, but it has not completely cooled down [which] according to the custom explained in the laws of cooking [which is] that it has become accustomed to be lenient [by precooked liquids even if it has cooled below Yad Soledes but is still warm], or [another example] even if [the food] has completely cooled down but it is a food which was baked or roasted [and is thus dry] [in which case we hold] that there is no [prohibition] to bake or roast an already baked or roasted food even if it has completely cooled down, [then in the above cases it is allowed to even boil these foods near the fire].
The prohibition to place it on or very near an open the fire: Nevertheless, [the Sages] only permitted [placing even precooked foods] opposite the bonfire however to place it on the fire itself, or even very close to the fire is forbidden. Rather one must distance it a little bit [from the fire].
The reason for this is: because there is suspicion that he may come to forget that it’s Shabbos and will come to stoke the coals. Requiring him to slightly distance it [from the fire] serves for him as a notice and reminder and he will no longer come to [forget and] stoke the coals.
If the fire is covered: [If the fire is covered, such as by a Blech or the like then one may place the cooked food even very near the fire even if the other Chazara conditions are not met, however it may not be placed on the fire unless all the Chazara restrictions are fulfilled.]
C. Placing food next to a Tanur and Kupach oven:
By a Tanur oven of back then and by a Kupach oven which has been fueled with olive waste or wood, one may never place food even near the oven on its outside, if that area is Yad Soledes.
D. Placing food near a heater:
To place food near a heater which is not swept or covered, even if it has fire in it, is permitted in all cases, as is our custom as explained above [in Halacha 20], as long as the food is [still] a little hot if it contains liquid, if it is Yad Soledes by the area that is next to the heater. However if it is not Yad Soledes there next to the heater then even if the food [which contains liquid] is completely cold, it is permitted to place it there in all cases, as explain above [in Halacha 21].
Summary-Placing food next to an oven by an area that is Yad Soledes:
This is never allowed if there is a prohibition of cooking involved in heating up the food, such as that the food has not been fully cooked, or has but it contains liquid which has completely cooled off. If there is no prohibition of cooking involved, such as that the food is fully cooked, and if it contains liquid the liquid is still warm, then the following is the ruling:
1. Tanur/Kupach: Is prohibited by a Tanur oven of back then, and by a Kupach which has been fueled with olive waste or wood.
2. Kirah oven: Is permitted by a Kirah oven according to all if the oven has been fueled with twigs and straw, or has had its ashes covered or removed. If however it was fueled with olive waste or wood, and its coals have not been removed or covered, then although the custom is to be lenient, it is proper to be stringent to not place food even near the oven, in a case that there is not so much need for one to do so.
3. An Open Flame/Stove Top: Food may not be placed literally right next to the open fire, as we are afraid that one may come to higher the flame. [However if the fire is covered such as by a blech, then it if it does not have a cooking prohibition, it is allowed to place it near the fire even initially on Shabbos, without fulfilling any of Chazara conditions.]
May one place food very close to his stove top fire?
If the fire is uncovered, then one may not place the food right next to the fire unless all the Chazara conditions are fulfilled. If the fire is covered by a Blech then seemingly one may place food even directly near the fire. 
 Admur 253:20
 See the The Laws of cooking
 The Sages
 Lit. Good
 Admur 318:24, Shabbos Kehalacha Vol. 1 page 423
 Admur 1:4
 So is evident from Halacha 4 above that all agree that when the fire in the Kirah is covered one may place the food next to the walls of the oven, and so is evident from Chapter 318:24 that only a bonfire, which is a revealed fire is mentioned in the restriction.
 See Rama 253:2 “By a Tanur there is no difference regarding returning on it or near it” [omitted from Admur regarding Chazarah]
Opinion of Admur: It is explained in Halacha 4 that even before Shabbos it is forbidden to place food even near the oven. Now, although there it was only referring to if the food was not cooked to the point of Ben Drusaiy, while here it refers to fully cooked food, nevertheless it is forbidden as is implied from Halacha 20 in which Admur specifies that the allowance to place food near an oven is by the Kirah oven and Tanur of today, and does not make mention of a greater novelty that it is even permitted by a Tanur oven. However, Tzaruch Iyun why Admur did not make mention in Halacha 15 that even to place a food near the Tanur is forbidden.
 Admur 253:27
 Meaning in all cases of whether or not one has intention etc. etc, however regarding the type of oven, it was already explained above in Halacha 20 that by a Tanur oven of back then it is always prohibited to place food even near the oven.
 So is evident from Halacha 5 above that all agree that when the fire in the Kirah is covered one may place the food next to the walls of the oven, and so is evident from Chapter 318:24 that only a bonfire, which is a revealed fire is mentioned in the restriction.