Touching the parchment of a Sefer Torah:
It is forbidden to hold [or even touch] the parchment of the Sefer Torah without a cloth. This applies even if one washed his hands. [This applies whether to the interior side of the parchment where the words are written, or to the exterior side of the parchment. However, with a cloth, such as with one’s Tallis, it may be touched and moved.]
Examples: During Hagbah, one may not hold onto the parchment of the Sefer Torah. Likewise, in any circumstance that the parchment of the Sefer Torah needs to be touched, such as if it became unaligned from the handle during Gelila, one is to use his Tallis to realign it. Likewise, when the Baal Korei shows the area of the reading to the Olah, he is not to touch the actual parchment but just point at the area with his finger from above, or use the silver Yad handle for this purpose. Likewise, a Sofer is to hold onto the Sefer Torah using a cloth if he needs to edit it during the reading.]
Not during Mitzvah times: Some Poskim rule that the above restriction only applies while one is performing a Mitzvah with the Sefer Torah, such as during its reading, or Hagbah and Gelila, however, during times that a Mitzvah is not being performed with it, such as one who moves the Sefer Torah from one area to another, or rolls it to the right place before the reading, or if a Sofer needs to edit it not within the midst of the reading, then there is no prohibition to hold it directly with one’s hands without a cloth. Practically, however, from the simple implication of the Poskim, it is implied that it is forbidden at all times, and so is the final ruling of Ashkenazim and many Sephardim.
In time of need: The above prohibition against touching the parchment of the Torah scroll without the use of a cloth, only applies when it is possible to accomplish whatever one desires to accomplish by using the cloth. If, however, one cannot accomplish the necessary action unless he directly touches the parchment, then it is permitted to be done. Thus, for example, if the Torah scroll needs to be fixed and re-sewed, it is permitted to touch the parchment for the sake of doing so. Nevertheless, it is proper for one to wash his hands beforehand.
 Michaber 147:1; Y.D. 282:4; Shabbos 14a “Rebbe Parnach said in the name of Rebbe Yochanon that whoever holds a Sefer Torah naked [i.e. bare handed] is buried naked without that Mitzvah.”; Megillah 32a; Miseches Sofrim 3:16; Kaf Hachaim 147:1-2
Other opinions: Some Poskim understand the above statement of the Talmud to be saying that one should not hold onto a Torah scroll if his upper body is not covered with a Tallis, and if he does so, then he is buried naked. [Maharit Mildula in Mayim Rabim 14:53; Hagahos Hatur 147; opinion mentioned in Mamar Mordechai 147:2, Kaf Hachaim 147:3] Practically, all the Poskim negate this understanding in the Talmud and rule as above. [Mamar Mordechai 147:2, Kaf Hachaim 147:3]
 M”B 147:1
 The reason: This is due to the honor of the Torah scroll, and that holding it with one’s hands can cause it to become dirty.
 Rama ibid; Darkei Moshe; Beis Yosef 147; Tosafus Shabbos 14b; Agudah Shabbos 20; Ritva Megillah 7b; Taz 147:1; Olas Tamid 147:2; Levush 147; Shaareiy Efraim 10:9; Aruch Hashulchan 147:1; M”B 147:1; Biur Halacha 147:1 “Vetov”
Other opinions: Some Poskim rule that it is permitted to touch the parchment of the Sefer Torah if one washed his hands before hand. [Beis Yosef 147 in name of Mordechai 4:834 in name of Ravayah 595, and that so was the Ashkenazi custom to hold onto the Sefer Torah parchment to show it during Hagbah; Implication of Rashba Shabbos 14a]
 Setimas Haposkim
 M”B 147:1
 Darkei Moshe 147:2 “I am from their descendants and never saw such a custom [in Ashkenaz to hold onto the parchment during Hagbah] and on the contrary, everyone is careful not to touch the Torah scroll parchment”; M”B 147:2
Other opinions: Some Poskim rule that the Ashkenazi custom was to hold onto the Sefer Torah parchment to show it during Hagbah [Beis Yosef 147]
 M”B 147:2; See Riva in Tosafus Shabbos ibid; Ben Ish Chaiy Toldos 18; Kaf Hachaim 147:1-2
 Kaf Hachaim 147:1
 Rav Shmuel Vital in Shaar Hakavanos p. 49b “not during times of the reading, such as when one needs to fix it, there is nothing wrong with holding”; Implication of Gemara ibid, as explained in Riva in Tosafus Shabbos ibid, that the entire issue is that one does not get reward at the time that one does the mitzvah if he holds the parchment, if however, he is not in the midst of a mitzvah, then it is not an issue; Ben Ish Chaiy Toldos 18; Kaf Hachaim 147:1; See wording of Riva brought in Taz 147:1 and M”B 147:1; Custom of Yemenite Jewry
 Setimas Kol Haposkim, Michaber, Rama, M”B who make no such mention of this leniency
 M”B 147:2 regarding that one may not touch the parchment to roll the scroll to the correct text before the Aliyah
 Rav Ovadia Yosef in Halichos Olam 3 p. 125
 P”M 147 M”Z 1; M”B 147:1
 Shaareiy Efraim 10:15; M”B 147:1