There is a passage in the Torah (Shemot 15:22-27) which describes Bnei Israel’s three-day journey in the desert without finding any source of water. Since water is compared to Torah, the Sages learn from this that three days should not go by without a public reading of the Torah. As such, the Gemara (Bava Kama 82a) explains that Moshe Rabbenu and Ezra HaSofer enacted that the Torah should be read on Monday and Thursday, and on Shabbat Minha, respectively. Part of this enactment was that three people should be called up and that a minimum of three verses be read per person. As is well known, if present, a Kohen is given the privilege of the first Aliyah and the Levi the second.
A situation arises when non-Kohanim need to be honored with first Aliyah, such as at a family celebration. Although the Kohen has the first right to the Aliyah, it is acceptable for him to forego this honor so that others may be called up to the Torah. The Rama (Orah Haim 135:1) says that a solution is to simply add supplementary Aliyot, such that the Kohen and the honorees may go up. Rabbi Yosef Benaim (Noheg BeHochma, pg.144) and Rabbi Moshe Toledano (HaShamaim Hadashim § 282) write that the Moroccan custom, however, is not to add Aliyot on the readings of Monday, Thursday or Shabbat Minha, but rather to ask the Kohen to temporarily step out of the sanctuary while the non-Kohanim are called up. This is not the case on Shabbat or the holidays, when it is common to add Mosifim, supplementary Aliyot. The HIDA (LeDavid Emet ch. 5) writes that one need not even ask the Kohen to step out, and that as long as he forwent his honor, he may stay in the sanctuary.
Summary: If necessary, a Kohen may be asked to temporarily leave the sanctuary so that a non–Kohen may get a first Aliyah when the Torah is read on Monday, Thursday or Shabbat Minha.