I've seen some people remove their Tefillin before Mussaf on weekdays of Rosh Chodesh. Why do they do this?
The Levush (brought by the Mishnah Berurah - Orach Chaim Siman 423) says that since the Teffilah of Mussaf is instead of the Korban Mussaf and when we give the Korban it is like a Yom Tov, therfore Just like on Yom Tov we do not wear Teffilin, so too, by Hallel we do not wear Teffilin. (As an aside he adds when the Kriyas Hatorah of the Korban Mussaf is read the Teffilin should also be removed.)
The Radvaz says the Minhag (custom) has no source, but the Minhag is mentioned in a Sefer called Elkana and that is why everyone does it. He continues and writes even though it is his personal practice he does not wish to write the reason and sends you to the Sefer Elkana for further research. He also says he is not ruling that anyone follow this practice. He then brings what he calls the simple answer, somewhat similar to the Levush since when you say Hallel it has a law of Yom Tov and Yom Tov is an Ois (sign) we do not need Teffilin as they are also an Ois (This is the stated reason we do not wear Teffilin on Shabbos and Yom Tov).
He then gives a Third reason. We know Teffilin are called Kesser (crown). Since we say Kesser in Mussaf there is no reason to have two of them and even if one is Daveninng by himself he should still remove it as it is the appropriate time to say the Kesser Tefillah.
The Taz quoting the Beis Yosef explains it somewhat differently he says since we are crowning Hashem it would be improper for us to be wearing a crown at the same time. The sefer Sohm Derech, by Reb Simcha Zissel Broide says this is similar to when all of the Kings at Yackov Avienu's funeral took off their own crowns and placed them on Yackov's coffin showing the superiority of Yackov to themselves. So too we remove our crowns to Show Superiority of Hashem to ourselves.