There is extensive evidence for ritual drama in Mesopotamia: myths and hymns written on clay tablets, ritual masks, and reliefs featuring festival processions. Rituals were performed by the king and members of the priesthood are were accompanied by music, singing, and dancing. One of the most important ritual dramas was the hieros gamos, or Sacred Marriage, which took place during the New Year Festival. It symbolized the union of the goddess Inanna/Ishtar and her lover Dumuzi. The Sacred Marriage was performed by the king in the role of Dumuzi and a priestess as Inanna, and likely included dialogues between the couple besides a physical union.
Another popular Mesopotamian ritual theme was the reenactment of a fight between a god and a monster. The Sumerian Epic of Gilgamesh features the adventures of Gilgamesh, a legendary king of Uruk, and his wild-man friend, Enkidu. One of the most famous stories is the battle between Gilgamesh and the monster Humbaba, also called Huwawa. Many masks, such as this one of Humbaba, have been found as well as masks depicting the hero character. Some of the earliest masks are dated to the early second millennium BCE.
Although no structures have been found which indicate a dedicated cultic theater for these dramas, Mesopotamian temples had large sanctuaries and courtyards where they may have been performed. The temples also had wide antechambers and courtyard benches where spectators may have been seated to observe the rituals.
Similar cultic rituals were performed in ancient Anatolia as well. The Hittites practiced the Sacred Marriage ritual between the Storm god Tarkhuna and the Sun goddess which was performed by the king and queen. They also recited the hero/monster drama during the Puruli festival, where Tarkhuna battles the dragon/snake monster, Illuyanka. Some texts included ritual directions which means the ceremony was probably regularly performed for the public. The Mother Goddess Cybele became the main deity in Anatolia by the first millennium BCE and was the subject of several cultic dramas. The Sacred Marriage was likely performed in honor of Cybele and her consort Attis, as well as a reenactment of the story of Attis’ death and Cybele’s resurrection of him.
During their festivals, Hittite cultic processions incorporated music and dance. They were led by the king and queen followed by dancers and even statues of sacred animals. Mimes also performed reenactments of a hunt disguised as bears or leopards.
The Hittites performed some rituals in their temple courtyards, which were similar to the Mesopotamian style, but rituals practiced in honor of Cybele were performed in caves and in special areas carved into mountainsides. A cave has been discovered that contained benches for spectators to sit and many lamps were found indicating some rituals were performed at night.
I think it is possible some of the rituals discussed such as the reenactments of the hunt could be considered theatrical drama since an audience was present and it may have been more entertaining than religious. However, the majority of these ritual dramas were too deeply entrenched in religion to evolve into theatrical drama.
 Inge Nelson, Cultic Theatres and Ritual Drama,(Oxford, Oakville, CT:Aarhus University Press, 2002), 39-40.
 Inge Nelson, Cultic Theatres and Ritual Drama, 41.
 Inge Nelson, Cultic Theatres and Ritual Drama, 47, 50.
 Inge Nelson, Cultic Theatres and Ritual Drama, 60-62.
 Stefano De Martino, “Music Dance, and Processions in Hittite Anatolia,” Civilizations of the Ancient Near East, Vol. IV, editor Jack Sasson, (New York: Charles Scribner’s Sons, 1995), 2666-2667.
 Inge Nelson, Cultic Theatres and Ritual Drama,(Oxford, Oakville, CT:Aarhus University Press, 2002), 63, 66.