Gemara

Even though one may sell and slaughter before the Goren, people (wait until tithing, for they) like to do Mitzvos with their property.

Chulin 110a (Rami bar Tamri citing Rav Yehudah): A borrowed garment is exempt from Tzitzis for 30 days.

Rishonim

Rosh (Chulin 8:26): R. Tam says that one should not bless on a borrowed Talis. Even though women may bless on Mitzvas Aseh sheha'Zeman Gerama, even though they are exempt, others (men) are commanded about them. Here, everyone is exempt. Even so, one who blesses does not lose. I say that the exemption of a borrowed Talis is if it did not have Tzitzis on it, the borrower need not attach Tzitzis. If one borrows a Talis with Tzitzis, it was in order to bless on it! If he can bless only if it is his, it is as if he gives it for a gift on condition that he will return it. We find that if one borrowed a garment and a relative died, he may not tear it, unless he told the lender that he is going to visit a mortally sick relative. Then, 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. If he found it folded, he must return it folded like it was. If not, the owner is not pleased.

Mordechai (Hilchos Ketanos (Menachos) 950): Semak says that one may bless on another's Talis, even if it was not given for a gift, as long as the owner gave permission. Even without permission, we assume that a person is pleased that others do a Mitzvah with his property. However, Maharam explained that if the Talis was left folded, he must ask permission. I think that if he will fold it like it was initially, it is permitted. Or, perhaps since the owner folded it, he revealed that he is particular not to lend 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. In a place where the custom is to leave Talesim in the Beis ha'Keneses, if one wants to wrap himself with one for a short time and return it to its place, it is permitted. He is not considered Sho'el she'Lo mi'Da'as (one who borrows without permission), who is a thief. Since there is no concern for consuming the principal, a person is happy that others do a Mitzvah with his property. However, the Tosafists say that one may not bless on a borrowed Talis. One should be concerned for their opinion.

Terumas ha'Deshen (100): The Rosh permits putting on another's Talis without his knowledge. We rely on the fact that one is happy that people do a Mitzvah with his property. This is when it is evident that there will be no loss to the property. If he found it folded, he must return it folded like it was. If not, the owner is not pleased. This is unlike Seforim, for they get dirty and ruined when one learns in them. The Gemara forbids learning in them Stam, i.e. even a little, for there no limit to how much one learns. Putting on a Talis or taking the Minim has a limit (it is for a short time).

Poskim

Shulchan Aruch (OC 14:4): One may take another's Talis and bless on it.

Beis Yosef (DH v'Rabbeinu): R. Yerucham says that some take Tzitzis that they find in the Beis ha'Keneses without the owner's knowledge. This is permitted. A person is happy to do a Mitzvah with his body or property. However, he must leave it folded, like he found it. If not, surely the owner is not pleased. He borrowed without permission; he is like a thief. His Berachah only angers Hash-m. It seems that the Rosh agrees. This is the custom.

Taz (6): The Yam Shel Shlomo (Chulin 8:53) says that one who borrows a Talis or Tefilin without the owner's permission may not take them outside the Beis ha'Keneses, for one does not want his property to be removed from his Chazakah, i.e. the place where he left them.

Magen Avraham (8): When one borrowed without the owner's permission, why may he bless? We must say that even though he is exempt, he may bless if he wants, like women bless on Mitzvas Aseh sheha'Zeman Gerama.

Magen Avraham (7): The Bach permits only on occasions, but not on a regular basis.

Mishnah Berurah (13): The Pri Megadim says that whenever the owner is there, it is proper to ask permission. We do not rely on the Chazakah when one can easily verify it, and all the more so if one knows that he is particular.

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

Kaf ha'Chayim (20): We are not lenient about Safek Berachos against a custom. (Rather, we follow the custom, and bless.) However, one who fears Shamayim should avoid this as much as possible, especially nowadays, for we see that most people are particular.

Kaf ha'Chayim (17): One may not borrow it on a Shabbos when everyone gathers to hear the Drashah and it is very crowded and it is prone to get damaged.

Kaf ha'Chayim (18): If the Talis became damaged, he must fix it. If he did not, he was not Yotzei.

Kaf ha'Chayim (16,27): If a Talis or Tefilin was left in a locked box, this shows that the owner does not want people to use it. One may not open it with a key to borrow it. The same applies to one who gives it to the Gabai to put away in a locked place. The Gabai may not lend it to another unless he clearly knows that the owner consents.

Taz (5): The Rema says similarly about a Lulav in Siman 649. The Terumas ha'Deshen equates this to a Talis. However, why do they permit only after the first day? Also Tzitzis is mid'Oraisa, and we rely (on the Chazakah that one is happy that people do a Mitzvah with his property) to bless on them without the owner's permission! I answer that there is no Isur Torah to wear a garment without Tzitzis, just there is a Mitzvah to attach Tzitzis on one's garment. There is no Chiyuv here, for it is not his garment. However, above (8:9) we are more stringent about the Berachah than about wearing a garment; one must check the Kashrus of the strings only due to the Berachah. Even though a borrowed Talis is exempt, we say that the owner transfers ownership to the borrower on condition to return it. The Rosh allows blessing on a borrowed Talis. One should be concerned for the Maharshal's opinion, and not rely on this, and all the more so if he borrowed it without the owner's permission. The Rosh said that the custom is to bless, but he also said that in the days of Selichos people borrow a Talis (for they begin at night when one may not bless on his own) and do not bless. The Rosh said that they err. I say that they are correct.

Shulchan Aruch (ibid.): However, he must fold it if he found it folded.

Magen Avraham (9): It is permitted even if he folds it differently than he found it, and on Shabbos, even if he does not fold it at all.

Mishnah Berurah (15): One should fold it like it was folded initially. On Shabbos one folds it unlike it was initially folded. The Magen Avraham is lenient on Shabbos not to fold it at all.

Kaf ha'Chayim (23): Keneses ha'Gedolah says that the owner is particular only that the borrower fold it, just like it was initially folded, but he need not fold it the same way it was folded. It seems that this is only for Shabbos, for then one may not fold it in the existing folds (302:3), but on a weekday, l'Chatchilah one must fold it like it was. Also, he did not rely on this reason by itself (see Kaf ha'Chayim 21 below). One should not be lenient like the Magen Avraham. The Pri Megadim and others disagreed.

Kaf ha'Chayim (21): In some places, when a Tzibur visits another Tzibur due to a Chasan or Bris Milah, they borrow Talesim and do not fold them afterwards. Indeed, the Nimukei Yosef connotes that one need not fold it afterwards, but Semak, the Mordechai, Rosh and R. Yerucham disagree. One should not rely on the Nimukei Yosef against them and enter a Safek Berachah l'Vatalah. It seems that since the Gabai later folds them, it is as if the borrower folded them. Also, since other times the 'host' Tzibur visits the 'guest' Tzibur, they are not particular. The custom of Yisrael is Torah. This is from Keneses ha'Gedolah. Beis Oved brought this, but gave only the first reason. It seems that Keneses ha'Gedolah required also the second reason, for sometimes the Gabai delays or forgets to fold them.

Kaf ha'Chayim (22): Even though one is happy that people do a Mitzvah with his property (even if he loses), this is only at the time of the Mitzvah. If one does not fold the Talis, it is damaged after the Mitzvah.

Kaf ha'Chayim (30): If one donated a Talis for the Shali'ach Tzibur, no one else may borrow it, even if he will not bless on it.