The Mishna establishes the well-known rule that whenever the Torah is read, the first Aliya must be given to a Kohen, the second to a Levi, and the subsequent Aliyot to Yisraelim. Some authorities claim that this constitutes a Torah obligation, but the consensus is that this sequence was established by the Sages.
The Shulhan Aruch (Orah Haim 135:4) rules that even if the only Kohen in the synagogue is ignorant in Torah, and an accomplished Torah scholar is present in the synagogue, the Kohen is given the first Aliya, as long as he is able to read along with the Hazan.
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) notes that this Halacha differs from the standard obligation of "Ve’kidashto," which requires giving a Kohen honor by granting him precedence (such as in leading a Zimun, and entering a room). Normally, a Kohen has the option of declining the honor and allowing somebody else, who is not a Kohen, to accept the honor. When it comes to the first Aliya to the Torah, however, a Kohen does not have this option. If there is only one Kohen in the synagogue, he must receive the first Aliya. This provision was instituted by the Sages to avoid conflicts in the synagogue. If the Kohen would be allowed to refuse the first Aliya and offer it to somebody else, other people may feel envious and resentful, leading to strife and contention among the congregants. Therefore, the Sages enacted that the first Aliya must be given to a Kohen, if the Kohen prefers declining the honor. Tosafot (Talmud commentaries by Medieval French and German scholars) comment that this applies even to the Torah reading on Monday and Thursday, despite the fact that on these days there is usually a smaller crowd than on Shabbat and holidays.
The Mishna Berura (135:12) addresses the question of whether this provision applies even when a Minyan is held in a private home. One could argue that since the Sages forbade a Kohen from refusing for the purpose of avoiding strife, this law should apply only in the synagogue, where people might argue with one another over the honor of the first Aliya. In a private home, however, the homeowner is recognized by all as the authority, and thus, perhaps, the Kohen should be allowed to refuse this honor and allow the homeowner to choose somebody else. The Mishna Berura concludes that since the Shulhan Aruch, in codifying this Halacha, draws no distinction between a Minyan in a synagogue and a Minyan in a private home, we should apply this Halacha even in private homes. By contrast, the Minhat Elazar (Rav Chaim Elazar Shapiro of Munkatch, 1871-1937) ruled that a Kohen may waive this honor if the Minyan is being held in a private home. Hacham David Yosef, in his Halacha Berura, cites both opinions, without issuing a definitive ruling on the subject.
If no Kohen in present in the synagogue, then the first Aliya should be given to a Torah scholar. The Mishna Berura, and Hacham David Yosef, note that a young Torah scholar even takes precedence over an older man who is not a Torah scholar.
It should be noted that the laws of precedence with regard to Aliyot are quite complex and intricate. Therefore, it is important for one who assumes the position of Mesader in the synagogue to review these Halachot to ensure that he gives precedence to the right congregants.
Summary: If there is only one Kohen in the synagogue, he may not refuse the first Aliya. According to some Halachic authorities, however, if a Minyan is held in a private home and not in the synagogue, and there is only one Kohen, he may refuse the first Aliya. If there is no Kohen present, the first Aliya should be given to a Torah scholar, even if he is young and there are others who are older than him.