Gemara

(Mishnah): The following are Posel Terumah [mid'Rabanan] ... a Sefer [Torah, Nevi'im or Kesuvim], [Stam] hands [that have not been guarded]...

Shabbos 14a (Beraisa): Even hands that touched a Sefer are Posel Terumah, due to R. Parnach's teaching:

(R. Parnach): One who holds a Sefer Torah bare will be buried bare.

Objection: This is unreasonable!

Correction: Rather, he will be buried bare of Mitzvos.

Objection: This is unreasonable!

Correction: Rather, he will be buried bare of that Mitzvah.

Question: Which of these two decrees was first?

We cannot say that the decree on Stam hands was first. If so, there would be no need to decree on hands that touched a Sefer!

Answer: They decreed first on hands that touched a Sefer.

Megilah 32a (R. Shefatyah): One who rolls the Sefer (closed) must put the seam in the middle.

Rishonim

Rif and Rosh (Megilah 11b and 4:12): One rolls the Sefer closed with the seam in the middle, so in case the Sefer will tear, the parchments will not rip, rather, they will come apart from one another.

Rambam (Hilchos She'ar Avos ha'Tum'ah 9:9): If cords or straps were sewn to a Sefer, even though one may not leave them [sewn], as long as they are attached, they are Metamei the hands.

Rambam (10): The case of a Sefer and cloths are Metamei the hands when they are sewn.

Tosfos (14a DH Kivan): It is not common to touch after washing and guarding his hands, so they would not decree just for this case.

Mordechai (Megilah 834): It seems that one may not hold a Sefer Torah bare with Stam hands, but if he washed his hands and was not Mesi'ach ha'Da'as (cease guarding them), we are not concerned.

Beis Yosef (DH v'Chasav): Perhaps Ashkenazim rely on this to hold the Sefer Torah bare when they lift it to show the writing to the people. However, I found in the name of the Agudah (Shabbos 19) that it is forbidden even right after washing.

Darchei Moshe (2): We descend from them (Ashkenazim), and we did not see this custom. We are careful not to touch the Sefer Torah bare.

Poskim

Shulchan Aruch (OC 147:1): One may not hold a Sefer Torah bare, without a cloth.

Magen Avraham (Reish ha'Siman): Perhaps the Mordechai holds that anything that cleans [the hands] suffices. The Bach and Maharit (1:136) forbid holding the poles directly. Ashkenazim are lenient about this. One should be stringent. The Rambam says that cords, straps, the case and cloths are Metamei the hands when they are sewn to the Sefer. Also R. Shimshon (3:5) says so from the Tosefta (2:5). If so, all the more so the poles [are Metamei the hands, so one may not hold them bare]. In Shabbos, we say that the primary reason they are Metamei the hands is so that one will not hold them bare! If so, one may not hold even the poles bare.

Kaf ha'Chayim (1): One who touches the Sefer Torah loses the Mitzvah he did at the time. This connotes that it is forbidden only when doing a Mitzvah, but not if he takes a Sefer Torah to put it somewhere, roll it to where they will read from it, or proofreads it. R. Shmuel Vital says that the Isur is only at the time of Keri'as ha'Torah b'Tzibur, for then great supreme lights are revealed. The Ben Ish Chai says that one must be careful when showing the writing to the Tzibur, and when rolling it during the reading. R. Shmuel Vital connotes unlike this, but it is good to be stringent.

Mishnah Berurah (1): One may not even merely touch a Sefer Torah bare without need, but not if there is a need, e.g. to sew a little or glue a cloth, etc. When there is no alternative, it is permitted. When there is no need, one should be stringent even after Netilas Yadayim, like the Rema concludes.

Mishnah Berurah (2): When they roll the Sefer Torah before reading, or during Hagbahah when the parchments sag and they hold the top of the parchments to straighten them, one must do so through a Talis or cloth. One may hold the poles directly. Some forbid, and wrap the poles in the end of the Talis at the time of Hagbahah. Where this is not the custom, one who wants to be meticulous about this should do so in a way that people not sense this, due to haughtiness.

Salmas Chaim (1:148): If one who ties the Sefer closed cannot avoid touching the parchment, it is not a problem.

Doleh u'Mashke (p.97): A Ba'al Korei should not point to where we will read if his finger will touch the parchment.

Rema: One may not touch a Sefer Torah even if he washed his hands.

Taz (1): The Rema forbids touching a Sefer Torah even right after washing. This is like the Agudah. Holding the poles, like our custom, is not considered holding it bare. The poles are only Tasmishei (things that serve a) Sefer Torah. Rashi says that the problem is to hold it without the cloth that is normally around it. The Gemara explicitly says that one must hold the poles at the time of the Berachah and reading! Chachamim did not say that one must take a cloth to avoid touching the poles. Why did the Beis Yosef discuss at the time of lifting the Torah, and not at the time of reading, which applies to all of Yisrael, and not only to Ashkenazim? (A Sefardi Sefer Torah is in a case.) Rather, the Isur is to touch the Sefer Torah itself, i.e. the parchments. Often, when they lift it, the parchments come down, and one holds the top of the parchment to pull it up, in order to roll it well. This is common among us (Ashkenazim). This is why the Beis Yosef said that Ashkenazim rely on the Mordechai, that washing permits this. The Rema forbids, i.e. touching the parchments themselves. Washing does not permit. There must be a Talis between his hand and the Sefer Torah. The Levush, Bach and Beis Yosef understood that the Gemara forbids touching the poles. I disagree. Even meticulous people are not careful about this.

Gra (DH v'Yesh): The Bach and Magen Avraham forbid touching even the poles bare, like R. Shimshon brings from the Tosefta. This is not a proof. Tefilin refutes the proof. They are Metamei the hands, yet Tosfos says that one may touch them. However, 'Sefer' connotes all Kisvei ha'Kodesh.

Kaf ha'Chayim (2): The Pri Chodosh understands the Bach to forbid even if the poles are wrapped in a cloth [for it is Batel to the pole - PF]. The hand must be wrapped. Many disagree. Since opinions argue, it is not haughtiness to be stringent. It is proper to do so!

Kaf ha'Chayim (3): The Beis Yosef says that the Ashkenazim rely on the Mordechai to hold the poles after washing the hands. The Beis Yosef himself says that we do not rely on this.

Kaf ha'Chayim (4): Some say that 'bare' means that the one holding it is not wearing a Talis, or he is naked. The Beis Yosef's opinion is primary, that it refers to the Sefer Torah. It is proper to be concerned also for the other opinions and have a Talis on his head for any Mitzvah involving a Sefer Torah. One does not refrain from the Mitzvah due to this.

Gra (DH v'Tov): The Gemara seems to support the Mordechai. It asked which was decreed first [and said that if they decreed on Stam hands first, there would be no need to decree on hands that touched a Sefer. If one may not touch even after washing, the latter decree would apply to one who touched after washing, because he transgressed]. It is necessary even after the first decree. However, Tosfos says that even after washing [one may not touch. The Gemara held that we would not decree due to this, for it is not common to touch after washing and guarding his hands.]

Shulchan Aruch (3): One who does Gelilah puts the seam in the middle, so if it will tear, the seam will tear.

Beis Yosef (DH v'Amar): Rashi says that he puts the seam in the middle, in order that he can close it tightly. The Rif (11b) explains that it is so that if it will tear, the seam will tear.

Kovetz Teshuvos (ha'Gaon R. Y.S. Elyashiv Ztz"l, 2:12, and Ha'Kri'ah b'Torah v'Hilchoseha 55(15) citing Sha'ar Efrayim 10:17). The Magbi'ah should open it with a seam in the middle even nowadays, when there is little concern lest the parchment tear.