Gemara

(Beraisa - R. Shimon ben Elazar): Every Mitzvah that Yisraelim were willing to die for when kingdoms decreed against it, like idolatry and Milah, they still solidly observe it;

Every Mitzvah that they were not willing to die for at the time of decrees, like Tefilin, their observance of it is weak.

(R. Yanai): One must [be able to] guard his body to wear Tefilin, like 'Elisha of the wings'.

Question: What does this mean?

Answer #1 (Abaye): He must be able to guard himself from passing gas [while wearing Tefilin].

Answer #2 (Rava): He must be able to guard himself from sleeping [while wearing Tefilin].

He is called 'Elisha of the wings' because Romi had decreed to kill anyone who will wear Tefilin. Elisha wore Tefilin in the market. An officer chased him and caught him. Elisha took the Tefilin from his head and put it in his hand. A miracle occurred. He opened his hand and revealed dove's wings.

Berachos 14b: Rav washed his hands, said Keri'as Shma, put on Tefilin and then prayed.

Contradiction: R. Chiya bar Ashi said that he often saw Rav wash his hands, bless, teach, put on Tefilin, and say Keri'as Shma!

Suggestion: Perhaps he would put on Tefilin before Shma only if the time for Shma had not yet come.

Question: In any case, there is a contradiction in Rav's practice!

Answer: The case was, the messenger delayed bringing Rav's Tefilin. (He needed to say Shma within the time, before his Tefilin came.)

(Ula): Reciting Shma without Tefilin is like testifying "I am a liar!" (He says in Shma that we must wear Tefilin, and he does not do so.)

(R. Chiya bar Aba): It is as if he offered an Olah or Shelamim without the Minchah or Nesachim

44b: In Eretz Yisrael, after removing Tefilin they bless "Asher Kidshanu b'Mitzvosav v'Tzivanu Lishmor Chukav."

Rosh Hashanah 17a (Rav): An example of 'Poshei Yisrael b'Gufam (sinners of Yisrael with their bodies)' is a head that never wore Tefilin.

Rishonim

Rosh (Hilchos Tefilin (after Menachos) 28): Since a miracle was done for Elisha with Tefilin, presumably he guarded them in Taharah. Nowadays, the custom is to wear them only at the time of Tefilah. It is easy to guard them then. Chachamim taught there (I found this only in Teshuvas ha'Ge'onim - PF) that on the day of judgment, if one was careful about the Mitzvah of Tefilin, the pan of merit outweighs. If one was negligent about them, the pan of guilt outweighs. There is no greater Mitzvas Aseh than Tefilin. Some people are lax about this, because Yisrael were not Moser Nefesh for it at the time of decrees, like it says in Shabbos.

Rosh (ibid.): The Yerushalmi says that Yisrael were not Machazik in Tefilin due to swindlers. Ploni was carrying money on Erev Shabbos. He found Almoni praying and wearing Tefilin. Ploni thought 'I should deposit only with people who fulfill Mitzvos like this!' After Shabbos, Almoni denied receiving the money. Ploni said 'I did not trust you, rather, in Hash-m's name on your head.' Do not say that Yisrael slackened in Tefilin because some Resha'im pretended to be Tzadikim and swindled people. Kesherim would not refrain from Tefilin due to this! Hash-m's ways are straight... (Tzadikim go in His ways, and Resha'im stumble in them.) Rather, the Yerushalmi asked why people do not establish those who wear Tefilin the entire day to be Kosher and trustworthy people. If one always has Hash-m's name on him, fear of Hash-m should stop him from sinning! Rather, it is due to swindlers, who wear Tefilin to deceive people.

Beis Yosef (OC 37 DH v'Chasav Od): Seemingly, the Rosh means like R. Yerucham explains, that people did not encourage Kosher people to wear Tefilin the entire day, due to swindlers, but at the time of Tefilah, any Yisrael who does not wear Tefilin is called Poshei Yisrael. However, scrutiny shows that the Rosh means like Tosfos (49a DH k'Elisha) says, that people did not establish those who wear Tefilin the entire day to be trustworthy, due to swindlers. Also the Terumas ha'Deshen (213) says so.

Rosh (Rosh Hashanah 1:5): The Rif's text says 'a head that never wore Tefilin.' We do not find such a text. Rather, it says 'a head that did not wear Tefilin.' R. Tam explains that he refrains because the Mitzvah is disgraceful in his eyes - "what benefit is there from these straps?!" We say that Tefilin requires a clean body, like Elisha. If one refrains because he fears that he cannot guard them in Kedushah and Taharah, even though he transgresses, for it is easy to be careful at the time of Keri'as Shema and Tefilah, he is not called ''Poshei Yisrael b'Gufam."

Ran (22b DH umid'Amar): Three Beraisos discuss sleeping with Tefilin (Sukah 26a). The Gemara concluded that one may sleep Arai when wearing Tefilin. The Ba'al ha'Ma'or says that we hold like this, even though the Rif omitted this. The Ra'avad says that we rely on R. Yanai, who says that Tefilin require a clean body, like Elisha, therefore one may not sleep in them at all. Letter of the law, one may sleep Arai, but we cannot gauge the time to walk 100 Amos, therefore we do not sleep with them at all.

Note: Surely, this is one 20th of the time to walk 2000 Amos (18 minutes; others say up to 24 minutes), i.e. about a minute. Rather, one who goes to sleep cannot gauge that he will sleep only this amount.

Tosfos (44b DH veli'Vnei): We do not bless after removing Tefilin. Bnei Eretz Yisrael removed them at night, for they held that the Torah forbids wearing them at night. We remove Tefilin in midday. Even if one removes them at night, we do not bless, for we hold that there is a Mitzvah at night, just we are concerned lest one pass gas.

Poskim

Shulchan Aruch (OC 37:2): The Mitzvah is to wear Tefilin the entire day. However, one must have a clean body and not emit gas, and not have Hesech ha'Da'as (divert his mind from them).

Beis Yosef (DH v'Chol): The Tur says 'one who does not wear them is called 'Poshei Yisrael b'Gufam.' Therefore, one must be very careful about them.' This seems unlike R. Tam. Rather, anyone who does not wear them, for any reason, is called 'Poshei Yisrael b'Gufam.' R. Tam says so only about one who refrains because he disgraces the Mitzvah.

Suggestion: The Tur means that Tefilin are so important, that there is a case in which one who does not wear them is called Poshei Yisrael, i.e. if he refrains due to disgrace. Therefore, one must be very careful.

Question: This is not true only about Tefilin. If one refrains from any Mitzvah due to disgrace, he is called Poshei Yisrael!

Answer: The Gemara mentioned Tefilin, and the same applies to other Mitzvos! However, perhaps Tefilin are different, for then he is called Poshei Yisrael b'Gufam, for one wears them next to the body, unlike other Mitzvos.

Rejection (of Suggestion - Beis Yosef): The Tur connotes that he holds unlike R. Tam. Whenever one refrains, he is called Poshei Yisrael. Also R. Yerucham says so. Also the Rosh, in Hilchos Tefilin, wrote Stam that Poshei Yisrael b'Gufam is a head that did not wear Tefilin.

Beis Yosef (DH v'Chasav Od): Tefilin do not require a Tzadik like Elisha, just that one can guard himself from gas, like Elisha.

Gra (DH veshe'Lo): The Gemara says in several places that one must constantly touch Tefilin [to avoid Hesech ha'Da'as]. The Tur and Shulchan Aruch omitted [the Isur to sleep in them], for this is included [in the Isurim of passing gas and Hesech ha'Da'as].

Bi'ur Halachah (DH Mitzvasan): The Pri Megadim was unsure if the Torah obligates wearing Tefilin the entire day, or just for one moment, and mid'Rabanan the Mitzvah is the entire day, and nowadays they were Mevatel this, for our bodies are not clean. He concludes that if one did not don Tefilin one day, he was Mevatel an Aseh. Through one moment, he fulfilled the Mitzvah, but the ideal Mitzvah is to wear them the entire day. Yeshu'os Yakov rules that the Torah Mitzvah is the entire day. Eliyahu Rabah proves from this that one who knows that he has a clean body should wear them the entire day. Clearly, he must be able to guard himself also from Hesech ha'Da'as, or at least from frivolity. Ma'aseh Rav says that if one is concerned for haughtiness among others, he may wear just the hand Tefilin. (It is covered, so others will not know that he is wearing Tefilin.) In any case it is better to make a small head Tefilin, and also the straps will be covered, than to go with only the hand Tefilin. Happy is one who fulfills what R. Zeira said, that he merited long life because he never went four Amos without Torah and Tefilin.

Shulchan Aruch (ibid.): Since not everyone can be careful about this, the custom is not to wear them the entire day. In any case, everyone must be careful to wear them at the time of Kri'as Shma and Tefilah.

Magen Avraham (2): The Shulchan Aruch connotes that even one who knows that he can guard the cleanliness of his body should not wear them the entire day, due to swindlers, like the Beis Yosef said in one Perush. The Rama of Pi'ano (Teshuvah 39) enacted to wear them at Minchah.

Kaf ha'Chayim (10): Also others say that it is proper to wear them at Minchah. The Ari Zal also prayed Minchah with Talis and Tefilin. At first he wore R. Tam Tefilin at Minchah. Afterwards, he made Tefilin according to Shimusha Raba (an early collection of Hilchos Tefilin) and wore them at Minchah.

Kaf ha'Chayim (11): We do not wear Tefilin at Minchah of Erev Shabbos or Erev Yom Tov. We wear at Minchah on Rosh Chodesh.

Mishnah Berurah (6): If one did not have Tefilin at the time of Keri'as Shema, or he had stomach illness at the time, the Mitzvah is the entire day. He must wear them, lest he be Batel one day from the Mitzvah of Tefilin.

Mishnah Brurah (7): People of great deeds learn after Tefilah while wearing Tefilin. They must be extra careful to avoid idle words. In addition to the Isur of idle words, through this one comes to Hesech ha'Da'as from Tefilin. See Ma'aseh Rav (18, in which the Gra encouraged wearing Tefilin the entire day).

Kaf ha'Chayim (8): The Ari Zal said, based on Kabalah, that the Mitzvah is primarily in the morning at the time of Tefilah. There is less of a Mitzvah and obligation the rest of the day.