Your question was addressed in Insights to the Daf, below.

===========================================================

Insights to the Daf Nazir 48

1) A NAZIR WHO IS A KOHEN GADOL

QUESTIONS: The Gemara discusses whether a Kohen Gadol who is also a Nazir may become Tamei in order to bury a Mes Mitzvah. TOSFOS (DH l'Aviv) and the Gemara in Zevachim (100a) discuss whether a Kohen Hedyot who is also a Nazir may become Tamei for a Mes Mitzvah.

(a) We know that there is a Halachic principle that "Ein Isur Chal Al Isur" -- when one Isur is already in force, a second Isur cannot take effect on the same item. Hence, if the person is already prohibited to become Tamei to a Mes because he is a Kohen Hedyot or Kohen Gadol, how could another Isur, the Isur of Nezirus, take effect on him to prohibit him to become Tamei to a Mes? (SHA'AGAS ARYEH #60)

(b) There is a Halachah that a Kohen Gadol is obligated to shave his hair once a week (Ta'anis 17a). However, a Nazir is prohibited from shaving his hair for at least thirty days. How, then, is it possible for a Kohen Gadol to become a Nazir? The Nezirus should not take effect, since it contradicts the obligations of the Kohen Gadol!

ANSWERS:

(a) The SHA'AGAS ARYEH answers that the Isur of Nezirus to become Tamei will take effect upon the Isur of a Kohen to become Tamei through the mechanism of "Isur Kolel." When the Isur of Nezirus takes effect, it includes more than just the Isur of Tum'ah; it includes an Isur to shave and an Isur to eat grape products. Hence, since it takes effect with regard to those other items, it also takes effect with regard to Tum'ah.

However, the AVNEI MILU'IM (Teshuvah 15) asks that, normally, two different types of Isur cannot constitute an "Isur Kolel." That is, an Isur must prohibit two of the same type of item in order to be considered an "Isur Kolel," such as an Isur which prohibits two items of food , or two forms of Tum'ah. An Isur which prohibits food and also prohibits Tum'ah (like the Isur of Nezirus) cannot be considered an "Isur Kolel."

The Avnei Milu'im therefore suggests that the "Isur Kolel" of a Kohen Hedyot who becomes a Nazir is that the Isur of Nezirus makes him prohibited to be Metamei even for close relatives (for whom he was permitted to be Metamei as a Kohen Hedyot). Since the Isur of Nezirus prohibits him from becoming Tamei from those relatives, we would think that it also prohibits him, through "Isur Kolel" to become Tamei to a Mes Mitzvah.

However, this does not explain why the Isur Tum'ah of Nezirus should take effect on a Kohen Gadol, because a Kohen Gadol is prohibited to be Metamei for close relatives even without the Isur of Nezirus! The HAGAHAH on the Avnei Milu'im writes that perhaps the Gemara's case of a Kohen Gadol who becomes a Nazir is referring to a Kohen who is not yet a Kohen Gadol who accepts upon himself to become a Nazir at the exact moment that he is appointed as Kohen Gadol. Hence, the Isur Tum'ah of Nezirus is able to take effect simultaneously with the Isur Tum'ah of a Kohen Gadol, since they take effect "b'Bas Achas" and it is not a case of "Isur Chal Al Isur."

The KEREN ORAH suggests another reason to explain why the Isur Tum'ah of Nezirus takes effect on the Isur Tum'ah of Kehunah. He refers the Gemara earlier (4a) which cites a verse that teaches that Nezirus is an exception to the rule of "Ein Isur Chal Al Isur." For this reason, the Gemara there says that the Isur of a Nazir to drink wine can take effect on a pre-existing Isur on the person to drink wine (such as when the person had made a Shevu'ah not to drink wine, according to Rebbi Shimon according to Tosfos there; see Insights there). Similarly, here, too, the Isur Tum'ah of a Nazir can take effect on the pre-existing Isur of a Kohen to become Tamei.

(b) The Mitzvah of a Kohen Gadol to shave each week is only a Mitzvah d'Rabanan. It is not a Mitzvah d'Oraisa nor a Halachah l'Moshe mi'Sinai. Therefore, the Isur d'Oraisa of a Nazir shaving overrides the Mitzvah d'Rabanan of a Kohen Gadol to shave each week.

Alternatively, according to the KEREN ORAH cited above, it is possible to learn the laws of the Isur of Tum'as Mes of a Nazir from the laws of the Isur of wine. Just like the Isur of Nezirus overrides a Mitzvah to drink wine (such as when the person had made a Shevu'ah to drink wine), so, too, the Isur of Nezirus overrides the Mitzvah of a Kohen Gadol to shave his hair once a week.

However, this answer will not work here, because when it comes to the Isur of Gilu'ach, there is a verse which teaches that a Gilu'ach of a Mitzvah (such as the Mitzvah of a Metzora to shave) does override the Isur of a Nazir to shave, in the case of a Nazir who becomes a Metzora. This is derived from the word "Rosho" in the verse discussing the Metzora's Mitzvah to shave (Vayikra 14:9). The Gemara earlier (44a) discusses the difference between a Mitzvah to do Gilu'ach, which does override the Isur of a Nazir to shave, and a Mitzvah to drink wine, which does not override the Isur of a Nazir to drink wine.

Therefore, if the Mitzvah for a Kohen Gadol to shave his hair each week is mid'Oraisa, then indeed a Nazir who is a Kohen Gadol will have to shave his hair each week (since the Mitzvah of the Kohen Gadol to shave overrides the Isur of a Nazir to shave), and at the end of his Nezirus he will have to step down from the position of Kohen Gadol in order to grow his hair for thirty days so that he can perform the Gilu'ach of a Nazir and bring his Korbanos Nezirus.