In presenting the Mitzvah of wearing Tefillin, the Torah states that we are to wear a written copy of the words of the appropriate Parshiyos describing Yetzias Mitzrayim on our hands and on our heads, in order that the Torah of Hashem will be on our lips (שמות י"ג:ט'). The Gemara in Shabbos (דף ק"ח.) derives from the seemingly extraneous phrase about the Torah of Hashem being on our lips that the material, such as the leather, used for writing Tefillin must be מן המותר בפיך, that is, from an animal whose meat could permissibly pass through our lips, namely, a kosher animal. The Gemara stresses, though, that the material must be only from a kosher species, but the particular animal need not be slaughtered according to Halacha in order for its skin to be used for Tefillin. In other words, as long as it is from a kosher species, the skin may be used even if technically the animal's meat could not be eaten for a certain reason. The Maharsha (שם בד"ה מלה"ד) explains that the requirement clearly is that only the species be kosher, since the whole discussion is about using the animal's skin, which is not edible anyway. The Gemara then cannot be saying that the material which is used for Tefillin must itself be able to be eaten; the Halacha therefore clearly means that the animal must be from a kosher species even though it may not be able to be eaten.
This Gemara seems to be discussing only the parchment on which the Parshiyos of the Tefillin are written. But what about other parts of the Tefillin, such as the Battim and the straps? Must they come as well only from a kosher species? And what about materials used for other Mitzvos? Is this Halacha limited only to Tefillin? The Gemara earlier in Shabbos (דף כ"ח:) quotes a statement that only the skin from a kosher species may be used במלאכת שמים, for spiritual activities in general, implying that this requirement extends to all Mitzvos, although the Gemara itself actually extends it only to other parts of Tefillin. The Rashba (חידושי הרשב"א לשבת שם בד"ה אלא) quotes from Rav Hai Gaon that in fact the whole requirement to use a kosher species is applicable to Tefillin only, which is where the Posuk is found, and he asserts that in the Mishkan, for example, they used skins from non-kosher animals. The Ran in Rosh Hashanah, however, (חידושי הר"ן לר"ה דף כ"ו: בד"ה ומיהו) discussing what animals a Shofar may come from, is not as convinced; he seems to lean towards the view that based on the above Gemara in Shabbos, a Shofar too must come only from a kosher animal, but he is not completely sure, and thus leaves it in doubt.
In the Shulchan Aruch (אורח חיים סימן תקפ"ו סעיף א'), the Ramo, based on this Ran, writes that a Shofar from a non-kosher animal is invalid. The Magen Avraham (שם ס"ק ג'), rejecting the view of those who say that the requirement to use products of kosher species is applicable only to Tefillin, explains that Tefillin is simply the paradigm for all Mitzvos of the Torah, a concept developed in the Gemara in Kiddushin (דף ל"ה.) based, in fact, on our very Posuk. Actually, there may be an even stronger proof from the Gemara in Makkos (דף י"א.) which indicates that even those who do not accept that Tefillin is a paradigm for other Mitzvos of the Torah in general, do admit that regarding the Halacha of using products only from kosher animals (מן המותר בפיך), other Mitzvos are compared to Tefillin. The Taz, however (או"ח סימן ל"ב ס"ק כ"ז) writes that even with regards to the other parts of Tefillin, such as the straps, the requirement to use leather from only a kosher species is a Halacha L'Moshe MiSinai, which implies that we cannot learn out anything about other Mitzvos from Tefillin.
The Noda BiYehudah (שו"ת נודע ביהודה מהדורא תניינא או"ח שאלה ג') presents a Teshuvah from his son discussing this topic, and he objects for several reasons to this view of the Magen Avraham that materials used for any Mitzvah must come from only kosher animals. First, he cites the beginning of the Sugya in the aforementioned Gemara in Shabbos (דף כ"ח.) which examines whether a certain animal whose skin was used for the coverings in the Mishkan was kosher or not. If, he reasons, the Magen Avraham is correct, there would have been no room for the Gemara to even consider the possibility that the animal was not kosher. He further points to the Gemara in Sukkah (דף כ"ג.) which indicates that an animal, if tied up, can be used as a wall for a Sukkah, and notes that the animal mentioned there is non-kosher, again posing a problem for the Magen Avraham. He also brings several proofs from the fact there apparently were certain dyes and fragrances used in the construction and the service in the Mishkan which were extracted from non-kosher species, including the Techeiles produced by the Chilazon, a non-kosher creature. It is noteworthy that elsewhere, the Magen Avraham himself (או"ח סימן ח' ס"ק ו') allows one to adorn a Tallis with material which comes from a non-kosher creature, although there is some discussion as to whether that material is in fact produced by the non-kosher creature or the shell in which the creature lives. In any case, the Noda BiYehudah's son asserts that the requirement to use material from only kosher species does not apply to all Mitzvos, but only to Tefillin and other Mitzvos which, like Tefillin, have to be written with Kedushah, such as a Sefer Torah or a Mezuzah. The Chasam Sofer (שו"ת חתם סופר חלק או"ח סימן ל"ט) likewise rules that only objects in the category of Tashmishei Kedushah, like Tefillin, Sifrei Torah, and Mezuzos, must be made from the products of kosher animals exclusively.
Interestingly, the aforementioned Ran regarding the Shofar can agree as well to this idea that Mitzvos in general do not need to be made from kosher animals; Shofar, however, may indeed require this because the Gemara in Rosh Hashanah (דף כ"ו.) states that the Mitzvah of Shofar is כלפנים דמי, indicating that it is equated to Avodah in the Beis HaMikdash and thus may have a different standard. The Chasam Sofer cited above asserts that this is the Ran's position; he then grapples with the question of why things used for a Shul, which presumably should be modeled after the Beis HaMikdash, are made of products from non-kosher animals. He answers that the Ran himself was unsure even about the Shofar, and thus rules that because of additional considerations, we can be lenient regarding a Shul. The Mishnah Berurah (או"ח סימן תקפ"ו ס"ק ח') seems to agree, writing that if one only has a Shofar from a non-kosher animal, one should certainly blow it on Rosh Hashanah, albeit without a Beracha because of this doubt.
The Torah Temimah raises another possible distinction, both on our Posuk (אות מ"ז) and later on (שמות כ"ה:ד',אות ד'), saying that only when the item for a Mitzvah must be made from an animal product must the animal be kosher. If, however, the item may be made of any material, and the person opts to use an animal product, it may come from a non-kosher species. The consensus of the Poskim seems to be that materials for most Mitzvos need not be made from the products of kosher animals only.