The Shulchan Aruch (141:1) rules that both the one who reads the Torah and the one called to the Torah must stand during the reading. It is forbidden for them to sit or even lean upon a wall or piece of furniture during the reading. The reason is that just as the Torah was given to Am Yisrael at Sinai amidst awe and trepidation, so must our Torah reading reflect the experience of awe. One must therefore ensure to stand fully erect as an expression of awe and reverence. The Mishna Berura (commentary to the Shulchan Aruch by Rabbi Yisrael Kagan, the "Chafetz Chayim," 1839-1933) adds that this Halacha applies even when one does not lean all his weight on the wall or piece of furniture. Even leaning casually, without supporting oneself entirely on the given object, is improper while reading the Torah or receiving an Aliyah.

Some people are unaware of this Halacha and lean upon the Teiva – the table upon which the Torah rests – during the Torah reading. This is incorrect; one must stand without leaning on any object while reading the Torah or when called for an Aliyah.

An elderly or otherwise frail person, who finds it difficult to stand erect without supporting himself, may lean on the Teba while he reads the Torah or is called for an Aliyah. Even such a person, however, must ensure not to lean on the cloth that covers the Teba and upon which the Torah rests. Since the Torah rests on that cloth, it has the status of "Tashmish Kedusha" – an article that directly serves an article of sanctity – and it may therefore not be used for personal, mundane purposes. Therefore, a person whom Halacha allows to lean while reading the Torah must first lift the cloth and lean directly on the wood of the Teba.

It is also permissible to lean on the Teba while reading the Torah if it is clearly done for the purpose of reading. The Torah scrolls of the Ashkenazim, for example, lay flat on the Teba, as opposed to the Sephardic scrolls which stand upright on the Teba. It is therefore necessary on occasions in Ashkenazic synagogues to lean forward against the Teba to read the top of the columns in the Torah scroll. Halacha allows leaning while reading or during one's Aliya for this purpose, as it is clear to all that the person leans for the purpose of reading, and not for comfort, and thus no irreverence of for the Torah reading is involved.

(Taken from Halacha Berura, p. 193 onward)

Summary: One may not sit or lean even part of one's weight against any object while reading the Torah or when called to the Torah. Somebody who feels frail may lean, but even in such a case one may not lean on the cloth upon which the Torah rests, and should instead lift the cloth and lean directly against the table. It is permissible to lean against the table if this is necessary for the purpose of reading the Torah, as is often the case in Ashkenazic communities, where the Torah scroll lays flat on the table.

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Standing While Reading the Torah or When Called for an Aliyah