A mourner who is observing Shib’a should not be called for an Aliya to the Torah, since it is forbidden for mourners to study Torah.
An interesting question arises in a case where a mourner who is a Kohen does not have a Minyan in his home, and prays in the synagogue, instead, where no other Kohanim are in attendance. Normally, if there is a Kohen present, he must be given the first Aliya to the Torah, as otherwise people might question whether he is in fact a legitimate Kohen. In light of this Halacha, should we perhaps allow a mourner to receive the first Aliya if he is the only Kohen, in the interest of protecting his reputation?
The answer is that even in such a case, the mourner should not receive an Aliya, and he does not need to be asked to leave, either. It can be assumed that the people in the congregation realize that this Kohen is in mourning and is therefore unable to receive an Aliya, and therefore, nobody will question his status as a valid Kohen. This case resembles a case where the only Kohen in the synagogue is in the middle of Shema or the Amida prayer when the Torah reading begins. Since he cannot be called for the first Aliya – as waiting for him to finish his prayer would inconvenience the congregation – and it is clear to everyone present that he cannot receive the Aliya because he is praying, and not because he is not a valid Kohen, he does not receive the Aliya. Likewise, if the only Kohen in the synagogue is a mourner, he does not receive an Aliya, because everybody knows the reason why he cannot be called to the Torah.
The exception to this rule is Shabbat, when a Kohen who is a mourner should be called for the first Aliya if no other Kohanim are present. Public displays of mourning are forbidden on Shabbat, and if the only Kohen in the synagogue does not receive an Aliya because he is in mourning, or if he is asked to leave so a Yisrael can be called, this would publicly display his status as a mourner. Therefore, if it happened on Shabbat that the only Kohen in the synagogue is a mourner, he should receive the first Aliya. This is the ruling of Hacham David Yosef, in his Halacha Berura.
Summary: If the only Kohen in the synagogue is a mourner observing Shib’a, the first Aliya is given to a Yisrael, unless this occurred on Shabbat, in which case the mourner should receive the first Aliya.