I'm thinking about theatre that exists to tell a story or something along those lines rather than strictly for religious or oral history purposes. Though if I'm misdefining theatre, please tell me.
There's widespread storytelling traditions, many of which contain theatrical elements. Many stories could only be told during the winter, such as Kiowa Saynday stories. Among Anishinaabe peoples, birchbark biting could be used to illuminate the stories being told; i.e. people create patterns by biting into sheets of birchbark with their canine teeth, then one can hold it up to the fire, which will shine through the holes.
Northeastern Coast tribes dance and act out stories during potlatches ceremonies, so this is probably the closest equivalent to Asian and European theatre; however, the dances and the potlatch are also ceremonial—not just for entertainment. The regalia worn by dancers at potlatch are absolutely phenomenal. For instance, carved transformation masks open to reveal different faces. Some elaborate raven masks have movable beaks.
The Hamasta dance of the Kwakwaka'wakw is famous/notorious, since the dancer acts out a cannibalistic ritual.
You can imagine how much Arctic peoples needed entertainment during winter months. Inuit, Yupik, and other northern people have finger masks, which enabled them to act out stories in enclosed winter lodges. The smiling figure represents a man, and the frowning figure represents a woman. The attached feathers and furs extenuate the dancer's movements.