Generally speaking, Halacha forbids calling two Kohanim to the Torah for consecutive Aliyot. If a Kohen is called for an Aliya to the Torah after another Kohen, people may cast aspersions as to the validity of the first Kohen, suspecting that his status as a Kohen has been called into question, thus requiring that another Kohen be called. The same applies to Levi'im: two Levi'im may not be called to the Torah for successive Aliyot, as this may give rise to some suspicions concerning the status of the first Levi.
Chacham Ovadia Yosef, in Chazon Ovadia (Hilchot Sukkot), establishes an important qualification to this rule. Namely, two Kohanim may be called consecutively to the Torah if there is some interruption in between the two Aliyot. For example, a Kohen may be called for Maftir even if a Kohen had been called for the final Aliya just beforhand. Since the Kaddish is recited before Maftir, thus causing an interruption between the two Aliyot, a second Kohen may be called for Maftir. Likewise, when a new Torah scroll is opened for an Aliya, it may be given to a Kohen even though a Kohen had received the previous Aliya. Closing one Sefer Torah and opening the other serves to separate one Aliya from the next, and they therefore may be given to two Kohanim.
It thus emerges that on Simhat Torah, three Kohanim may be called to the Torah consecutively for the Aliyot of Chatan Torah, Chatan Bereishit, and Maftir. These three Aliyot involve readings from three different Sifrei Torah, and hence calling Kohanim for all three does not violate the prohibition against calling Kohanim for consecutive Aliyot.
Summary: A congregation may not call two Kohanim or two Leviim for consecutive Aliyot, unless the two Aliyot are interrupted by a Kaddish or the second Aliya is read from a new Sefer Torah.
