Removing the Tefillin before Musaf:
The custom is to remove the Tefillin prior to the silent prayer of Musaf. This applies even in those communities that recite Nakadishach in Musaf. [This applies even for one who is Davening in private. According to Kabala, there is a grave prohibition against Davening Musaf with Tefillin.]
What is one to do if he is still wearing Tefillin by Kedusha of Musaf?
One is to remove the Tefillin prior to the Chazan reaching Keser. If one is unable to do so, such as he suddenly realized that the Chazan reached Keser, he is to move his Shel Rosh to the side of his head and place his Tallis, or a cloth, under his Shel Yad, thus making an interval between it and his arm. Alternatively, if one does not want to answer Kedusha, he is to leave to another room. Some Poskim rule that if one is prior to Shacharis, or prior to completing the prayer of Shacharis, one need not be particular against reciting Keser with the Tefillin. Many are accustomed to simply cover their Tefillin [and remain wearing them] upon the recital of Keser if they are not Davening with this Minyan of Musaf.
Must one repeat the Davening if he accidentally Davened Musaf with Tefillin?
Remembered in middle of Shemoneh Esrei: In the event that one forgot to remove his Tefillin and remembered in middle of Musaf, he is not to stop Davening Musaf in order to remove them. Nonetheless, he is to move the Tefillin to the side, outside of their valid parameter.
May one wear Tefillin after Musaf?
Some Poskim rule one may wear Tefillin after Musaf without limitation. Other Poskim rule one may not wear the Tefillin until Mincha. Others rule one may not wear Tefillin anymore that day. Practically, if one has not yet worn Tefillin that day he may wear the Tefillin immediately after Musaf.
 Michaber 423:4; Rama 25:13; Admur 25:41; Siddur Admur
On Rosh Chodesh one is required to remove the Tefillin prior to the silent prayer of Musaf. [Michaber ibid] This applies in those communities that recite the Kedusha of Keser in Musaf, in which case the Tefillin is to be removed before Musaf, after having said the Kedusha of Uva Letzion, being that it is improper to be wearing Tefillin while saying Keser. Furthermore, even those communities who do not say Kedushas Keser in Musaf [but rather say Nakdishach] are accustomed to remove the Tefillin prior to Musaf. [Admur ibid; Rama ibid]
 The reason: As it is improper to be crowned with Tefillin while saying “Keser Yitnu Lecha”. [Admur ibid; M”A 25:32; Beis Yosef] Alternatively, the recital of Musaf is in place of the Karban Musaf, and during the time of the Musaf offering it is considered a Yom Tov, and Yom Tov contains an Os which is exempt from Tefillin. [Levush 25:13; 423; M”B 423:4; Kaf Hachaim 423:26] There is also a Kabalistic reason mentioned behind this custom. [Shaar Hakavanos p. 38; Kaf Hachaim 423:26]
 Admur ibid and Rama ibid that so is the custom; See Background
 See Zekan Ahron 2:5; Piskeiy Teshuvos 25:28
 Rashal in Yam Shel Shlomo Beitza 1:51; Bach 26 in name of Iggur, in name of Zohar, based on Midrash Hanelam Shir Hashirim that one who Davens with Tefillin during Musaf of Rosh Chodesh and Chol Hamoed is liable for death.
There are various opinions recorded regarding when the Tefillin are to be removed before Musaf. [See other opinions] The Alter Rebbe writes in the Siddur that the Tefillin are to be removed prior to the Kaddish recited before Musaf. In Peri Eitz Chaim however it is brought that the Arizal would remove the Tefillin after the Kaddish said before Musaf. The Shaar Hakavanos similarly says that this is the custom, and so is brought in other sources. Accordingly, this ruling of Admur requires analysis, as the entire custom of when to remove the Tefillin is based on Kabala and who can argue with the Arizal in regards to Kabala? The explanation is as follows: According to the Alter Rebbe one needs to hear three Kaddeishim with Tefillin prior to removing them. Thus, by a regular weekday Shacharis, one must wear the Tefillin until after the Kaddish recited after Uva Letziyon. [Evidently the Kaddish said before Hodu does not count as part of these three Kaddeishim. Some suggest that this is because this Kaddish is not always recited. Vetzaruch Iyun. The Kaddish recited after the Torah reading is not included in the three Kaddeishim being that it is not relevant to the prayers.] However, on Rosh Chodesh, being that according to Admur the Shir Shel Yom is said before Uvah Letziyon, with a Kaddish that follows it, one has thus already heard three Kadeishim before Uva Letziyon and does not need to hear the Kaddish of after Uvah Letziyon while still wearing Tefillin. Based on this, Admur rules that the Tefillin are to be specifically removed before the Kaddish, being that this Kaddish really belongs to Musaf, of which is to be said without Tefillin, and hence if one is not required to wear the Tefillin for this Kaddish he should specifically remove them beforehand. However, according to the Arizal, it seems that Shir Shel Yom was not said on Rosh Chodesh, or was said after Musaf, and therefore the Kaddish said before Musaf was still part of the three required Kadeishim. Now, being that the positioning of Shir Shel Yom is a dispute in the revealed plains of Torah, it does not form an issue for the Alter Rebbe to disagree with the Arizal on when the Shir Shel Yom should be said, and consequently on when the Tefillin are to be removed, which in it of itself is a matter of Kabala. According to the above it is understood that the ruling brought in the Siddur Yaavetz to remove the Tefillin after Kaddish, despite that one is to say the Shir Shel Yom after Shacharis, before Musaf, is a contradiction. [Igros Kodesh 2:50, printed in Shulchan Menachem 2:195 and Shaareiy Halacha Uminhag 1:176; Ketzos Hashulchan 8 footnote 56]
 Admur 25:41; Siddur Admur; M”A 25:30; Elya Raba 25:25
The reason: The reason for this is because one needs to hear four Kedushas with his Tefillin and thus the Tefillin are worn until after one hears four Kedushas, as Barchu is also considered a Kedusha. [M”A 423:6]
Other opinions: Some Poskim rule the Tefillin are to be removed during Uvah Letziyon. [Peri Megadim brought in M”B 423:10]
 Siddur Admur; Elya Raba 25:25, brought in M”B 25:59; Ketzos Hashulchan 8:16 footnote 56; Os Chaim Veshalom 25:19; Igros Kodesh 2:50 [printed in Shulchan Menachem 2:195 and Shaareiy Halacha Uminhag 1:176]
The reason: The reason for this is because the half Kaddish relates to Musaf and by Musaf we do not wear Tefillin. [Rebbe ibid]
Other opinions: Some Poskim rule the Tefillin are to be removed after the Kaddish of before Musaf. [M”A 423:6; Arizal in Peri Eitz Chaim and Shaar Hakavanos, bought in Shaareiy Teshuvah 25:13 and Kaf Hachaim 25:94-95; Siddur Yaavetz] The reason for this is because one needs to hear four Kedushas and three Kaddishim with his Tefillin and thus the Tefillin are worn until after one hears three Kadeishim and one Kedusha. [M”A 423:6] However according to the Nussach of Admur that there are three Kaddeishim heard before Uva Letziyon, it is not necessary to wear the Tefillin during this Kaddish. [Ketzos Hashulchan 8 footnote 56; Rebbe ibid] See Background!
 Ben Ish Chaiy Vayikra 17; Torah Leshma; Kaf Hachaim 25:99; Piskeiy Teshuvos 25:28
 Piskeiy Teshuvos ibid
 Shulchan Hatahor 25:13; Ashel Avraham Butchach 25
 The reason: As one is still prior to receiving the Kedusha of Musaf and Yom Tov, and hence his Tefillin do not contradict the Keser being heard. [ibid] Perhaps, based on this, people who are in middle of Shacharis are accustomed to cover the Tefillin upon the Chazan reaching Musaf, but do not remove it. However see Admur ibid that one is not to recite the words Keser while wearing Tefillin, irrelevant of the concept of the Kedusha of Yom Tov. Vetzaruch Iyun!
 Miaseif Lechol Hamachanos 25:136; See previous footnote!
 Kaf Hachaim 25:98
 M”B 25:61
 Piskeiy Teshuvos 25 footnote 215
 Rameh Mipuno 108; Chida in Machazik Bracha 25:13 and Moreh Baetzba 10:181, brought in Shaareiy Teshuvah 25 [Kaf Hachaim ibid concludes that if the Chida would have seen the ruling of the Rashash he would have retracted his ruling]; Mishmeres Shalom 30; Artzos Hachaim and P”M brought in M”B 25:60
 Rashash in Nehar Shalom p. 14; Ben ish Chaiy Vayikra 18; Kaf Hachaim 25:6-7; Imrei Pinchas
 Glosses in Peri Eitz Chaim that so was the custom of Rav Chaim Vital, brought in Or Tzaddikim 9; Shulchan Hatahor 25:12; Daas Torah 25 The Kaf Hachaim ibid negates the authenticity of this claim.
 Mishmeres Shalom 30; Os Chaim Veshalom 25 footnote 20