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It is customary of all Jewry to Kasher counters and tables that are used with hot foods during the year. Likewise, tables which are used without a tablecloth during the year, or are used with a thin cotton/linen/polyester tablecloth, are to be Kashered. However, tables that are only used with a thick or plastic/PVC tablecloth during the year, from the letter of the law do not need to be Kashered.
How to Kasher: Counters and tables are to be Kashered with Iruiy Keli Rishon and Even Meluban.
Is my table/counter a Kasherable material? Marble counters, and wooden tables are Kasherable. Quartz counters [i.e. engineered stone] are not Kasherable. Plastic tables are disputed if they are Kasherable. [See Halacha 8!] Practically, in all cases that the counter/table cannot be Kashered one is to nevertheless Kasher them with Iruiy Keli Rishon and then cover them with tinfoil, or PVC, as explained next.
Covering the counter or table: If one covers his table or counter with a thick tablecloth, PVC, or tinfoil, then from the letter of the law it does not need to be Kashered, as explained in Halacha 5. Likewise, if the counter or table has been Kashered, then from the letter of the law it does not need to be covered. Nonetheless, the custom of Jewry is to do both; to Kasher the table/counter and then cover the surface with tinfoil or PVC, as explained in the Q&A.
The walls of the counter: The walls surrounding the counters are to have Iruiy Keli Rishon performed and are then to be covered.
Must one cover all kitchen surfaces such as tables, counters, cabinets, refrigerator shelves and the like?
From the letter of the law, once these areas have been properly cleaned and Kashered they may be used for all foods without any cover. However, some Poskim rule that one is to cover the surfaces even after they are Kashered due to suspicion that perhaps they still contain actual Chametz that was not properly removed. Practically, the widespread custom is to cover all items that contact food even after they have been cleaned and Kashered.
 Admur 451:58 regarding tables; Piskeiy Teshuvos 451:44
 The reason: As at times people place a hot pot on them, and the pot spills, thus requiring Kashering of Iruiy Keli Rishon [according to the Poskim who rule we follow minority usage]. Furthermore, at times, people place actual hot Chametz directly on the counter, and it hence requires Hagala, or Iruiy Keli Rishon with Even Meluban. [Admur ibid] Nonetheless, Bedieved, if one placed a hot pot/food directly on the table or counter without Kashering it, everything remains Kosher as we follow majority usage by a non Ben Yomo item. [Admur 451:27 and 72]
 See Admur ibid that if it is not common to place hot foods on them, then Kashering is not necessary; See Halacha 5 for the law of a covering and the differences between materials!
 Michaber 451:20; Admur 451:58
 M”A 451:38 in name of Maharil [even regarding tables]; Elya Raba 451:40 [only regarding shelves, not tables]; M”B 451:115 [brings both opinions]; Dvar Moshe Tinyana 98:36; See Nitei Gavriel 77:1 footnote 2
Does this law apply even to tables and counters or only to shelves? The M”A ibid applies the ruling of the Maharil to all surfaces that contact food, including tables. However the Elya Raba ibid limits this stringency of the Maharil to cabinets with shelves in which it is difficult to clean the corners of Chametz, and hence it is to be covered. However, a flat surface such as a table, even the Maharil agrees that there is no need at all to cover it. The M”B ibid brings both opinions and seems to side with the ruling of the Elya Raba. So also rules the Aruch Hashulchan 451:41. Nevertheless, it is customary of Jews, including Anash, to be stringent in this matter.
Opinion of Admur: Admur ibid completely omitted the ruling of the Maharil; M”A and Elya Raba. He only mentions the necessity to cover an item in 451:48 regarding vessels made of wood that are constantly used with dough and flour; and in 451:65 regarding an oven that was not Kashered. No mention is made anywhere regarding covering an already Kashered table or cabinet. Nevertheless, it is customary of Jews, including Anash, to be stringent in this matter. [See Nitei Gavriel ibid]
 M”A ibid; Piskeiy Teshuvos 451:44