Gemara

(Beraisa): All agree that if on Shabbos one put Tzitzis on his garment or fixed a Mezuzah in his doorway, he is Chayav.

131b (Rav Nachman): This is because he could make his garment or house Hefker. (Since the Mitzvah is not obligatory, it does not override Shabbos.)

Chulin 136a: "Kesuscha" (your garment) teaches Rav Yehudah's law, that a borrowed garment is exempt from Tzitzis.

Rishonim

Rosh (Hilchos Tzitzis 1,2): In Ashkenaz, the custom is that before Yom Kipur, when they rise early [for Selichos], the Shali'ach Tzibur borrows another's Talis to exempt himself from a Berachah, since it is a Safek whether he should bless on his own [before one can see a person four Amos away]. They err. We exempt a borrowed Talis without Tzitzis, i.e. he need not attach Tzitzis until 30 days. One who borrows a Talis with Tzitzis must bless. Ba'al ha'Itur says so.

Beis Yosef (DH veha'Rosh): The Rosh is correct. This is the custom.

Rosh (Chulin 8:26): R. Tam says not to bless on a borrowed Talis. Women may bless on Mitzvas Aseh sheha'Zeman Gerama, even though they are exempt, for others (men) are obligated. Here, everyone is exempt. Even so, one who blesses does not lose. We exempt only a borrowed Talis without Tzitzis. If one borrows a Talis with Tzitzis, it is in order to bless on it! If he can bless only if it is his, it is like a gift on condition that he will return it. We find that if one borrowed a garment and told the lender that he is going to visit a mortally sick relative, it is as if he authorized him to tear it [and pay for the loss to the garment]. The custom is to put on another's Talis without informing him, and bless on it. A person is pleased that others do a Mitzvah with his property.

Yam Shel Shlomo (Chulin 8:53): Women are exempt from Mitzvos like Sukah, but others (men) are obligated, and they cannot evade it. A borrowed Talis is different. Everyone can evade the Mitzvah of Tzitzis or Mezuzah, if he does not have a four-cornered garment or a house. If so, a borrower should not bless. One who borrows to tear cannot avoid tearing, for he cannot go [to visit the Choleh] naked. It is as if the lender transferred ownership to him for that matter (tearing). One need not wear a Talis for Tzitzis. However, the custom is to bless on a borrowed Talis, for Yisrael accepted not to be without Tzitzis, just like Tefilin. Even though it is not a total obligation, since Hash-m commanded it to remind us [of the Mitzvos], He wants us to wear a four-cornered garment, to arouse our memory. Hash-m punishes [for not wearing Tzitzis] when He is angry (Menachos 41a). Therefore, the lender intends that the borrower acquire, to be Yotzei his obligation. It is as if it is his. However, one who fears Shamayim should not wear another's Talis. If there is no other Talis, he wears it without a Berachah, for the Berachah is not essential. Even those who say to bless, this is only for a Talis worn at the time of Tefilah, with Tefilin. If one borrows a Talis for an Aliyah, or to be Shali'ach Tzibur, he does not bless.

Rosh (Makos 1:4): A borrowed Talis is exempt from Tzitzis for 30 days. Since it is rare to lend for more than 30 days; we obligate it, lest people think that he bought it.

Nimukei Yosef (Hilchos Ketanos 12a DH Talis, citing the Ritva): A borrowed Talis is exempt from Tzitzis. If one wants to put Tzitzis on it and bless, this is praiseworthy. The custom is to bless on a borrowed Talis. My Rebbi says so. The Tosafists say that one may not bless on a borrowed Talis. One should be concerned for their opinion.

Poskim

Shulchan Aruch (OC 14:3): If one borrowed a Talis without Tzitzis, he is exempt from putting Tzitzis for 30 days. "Kesuscha" excludes others' garments. After 30 days it is obligated mid'Rabanan, for it looks like his.

Beis Yosef (DH v'Chasvu): Tosfos and the Rosh say that it is good not to bless on a borrowed Talis. Even so, one who blesses does not lose. Semag and Semak agreed.

Magen Avraham (5): Tosfos and the Rosh say that if one put Tzitzis on a borrowed garment and blesses, he does not lose.

Mishnah Berurah (9): One may put on Tzitzis and bless, just like women bless on Mitzvos they are exempt from. Therefore, one may go in Reshus ha'Rabim with it on Shabbos, for the Tzitzis are an adornment for the garment.

Kaf ha'Chayim (14): We are lenient about Safek Berachos, even against a majority and against the Shulchan Aruch. Therefore, one never blesses on a borrowed Talis. However, the custom is to bless; this overrides concern for Safek Berachos. However, perhaps this is only when borrowing for Tefilah, when he dons Tefilin. In some places, the custom is to bless even for an Aliyah or to Duchan (give Birkas Kohanim). We do not protest against them. A custom must be based on Chachamim.

Rema: If he returned it within 30 days, and borrowed it again, the days do not join. It must be 30 consecutive days.

Gra (DH v'Im): If he returned it in the middle, it does not look like his.

Mishnah Berurah (10): One who rents is like one who borrows (Tosfos Menachos 44a DH Talis). Ba'al ha'Itur was unsure.

Rema (ibid.): If one borrowed it with Tzitzis on it, he blesses on it immediately.

Taz (4): This is like the Rosh. I say that he discusses a Talis special for the Mitzvah, like Talesim of the Beis ha'Keneses, or a Talis Katan. However, if people wear a garment like clothes, just they put Tzitzis on because it has four corners, a borrower does not bless. Perhaps the lender lent it only for the sake of wearing it, we do not add Aliyos for the Mitzvah! This is like a Talis [without Tzitzis] that one borrowed from others. The borrower need not attach Tzitzis. He lent it solely for the sake of wearing it. (If he lent it for the Mitzvah, the borrower would acquire, and be obligated to attach Tzitzis, like on his own garment.)

Taz (4): Also I question the Rosh's proof. There, the garment was lent in order that he may tear it. Even if one lends a Talis in order that the borrower may bless, we cannot learn from tearing that the lender intends for a gift [on condition to return it].

Magen Avraham (6): The Rosh (Reish Hilchos Tzitzis) connotes that a Shali'ach Tzibur blesses on the Talis. Piskei Tosfos (Menachos 161)) says that if a Kohen wears a Talis to Duchan, he blesses on it.

Gra (DH Sha'alah): We say that one who lends an Esrog gives it in a way that the borrower is Yotzei. If not, what was the point of the gift?! (Bava Basra 137b)

Mishnah Berurah (11): Some say not to bless unless he explicitly gave a gift [on condition ...] Most Acharonim hold like the Shulchan Aruch. Surely it is best to explicitly ask for a gift. If one borrows a Talis for an Aliyah, to be Shali'ach Tzibur or Duchan, which is merely for honor, he does not bless. Some say to bless in any case. Derech ha'Chayim says that it is best to intend not to acquire it. Then, all exempt from blessing, unless he borrowed it for the Tefilah, to be Yotzei through it. This refers to private Talesim. All obligate blessing on Talesim of the Beis ha'Keneses, even just to be Shali'ach Tzibur or for an Aliyah, since they are bought with intent that it belong to the one who wears it, just like an Esrog [of the Tzibur].

Bi'ur Halachah (DH Sha'alah): Why must we say that they buy Talesim of the Beis ha'Keneses with intent that it belong to the wearer? In any case, Eliyahu Rabah says that it is like a Talis of partners, which is obligated! The Mordechai (Hilchos Tzitzis 950) says so; we need not say that partners in an Esrog intend that the one who shakes it owns it at the time. Perhaps Derech ha'Chayim holds that in a big city, everyone's share of the Talis is less than a Perutah.

Bi'ur Halachah (ibid., citing Sha'arei Efrayim): For an Aliyah, it is better to borrow a friend's Talis than the Tzibur's Talis, to avoid the Safek Berachah. If one wants to bless on a Talis of the Tzibur, he should intend to wear it for a garment of Tzitzis, and wrap his head in it. He need not wait the time to walk four Amos. If he does not want to bless, he wraps only his body, but not his head. A minority of Poskim obligate blessing even for wrapping the body, but here he intends just for honor. Such a wearing does not obligate a Berachah even regarding his own Talis. If pressed for time, one may wear it on the shoulders for an Aliyah without a Berachah.

Bi'ur Halachah (ibid.): The Shulchan Aruch (8:2) concludes that wrapping the head is not Me'akev. The Maharshal, Magen Avraham and Eliyahu Rabah always obligate blessing on his own garment. However, intent to wear for the Mitzvah is Me'akev, for we hold that Mitzvos require intent, and this is Me'akev (60:4).

Note: Ha'Gaon R. C. Kanievsky Shlita (personal communication) says not to bless on a public Talis in a Yeshiva [for one is not a partner in it, unless the Rosh Yeshiva decides to let people acquire it in order to bless on it.]

Mishnah Berurah (12): One must return the Talis before Ploni (the original owner) needs it, and he must intend to give it back for a gift, so Ploni will own it.

Shulchan Aruch (4): One may take another's Talis and bless on it.

Magen Avraham (8): Without the owner's permission, he is exempt. Even so, he may bless, like women bless on Mitzvas Aseh sheha'Zeman Gerama.

Mishnah Berurah (14): Derech ha'Chayim says that it is better to intend not to acquire it, and not bless.

Taz (5): The Rosh allows blessing on a borrowed Talis. One should be concerned for the Maharshal, and not rely on this, and all the more so if he did not ask the owner. The Rosh said that those who borrow a Talis for Selichos err. I say that they are correct.

Gra (YD 286:19): In Shabbos we say that one could make a Talis Hefker, i.e. one is exempt even if he [borrows and] wears it.