Many of the stories from the Tale of the Heike appear in later narrative and dramatic traditions. In narrative, other war tales, including Gikeiki (The Record of Yoshitsune) and Soga monogatari (The Revenge of the Soga Brothers) are based on the events of the Tale of the Heike and involve some of the same characters. Numerous shorter stories, some of them originally part of oral storytelling traditions, also concern events and characters from the Tale of the Heike. The influence of the tale is particularly strong, however, in the dramatic traditions evolving during the medieval period, including kōwaka-mai, noh, and jōruri, the story-telling tradition from which the early modern arts of bunraku (puppet theater) and kabuki emerged. Kōwaka-mai and noh developed concurrently with the popularity of the Tale of the Heike. Kōwaka-mai is a dramatic form with strong narrative tendencies. Although we don't know what it looked like in the medieval period, texts that probably record the tradition from that period demonstrate that it has a very strong narrative element: the libretto are generally in the third person and very closely resemble a source text from the military tale tradition, often the Tale of the Heike. Today, kōwaka-mai is only preserved (or resurrected) in small communities, most famously Ōe village in Fukuoka (Kyushu), where it is performed annually shortly after New Years and has been designated an Intangible Folk Cultural Asset. You can see a short television report featuring it here: http://webtv.pref.fukuoka.lg.jp/en/movies/detail/1158.
The noh drama developed in the late 14th and early 15th century; the father and son team of Kan'ami and Zeami, two generations of leaders of the Yuzaki troupe, are credited today as its founding fathers, although the breadth of what constituted noh during the medieval period encompassed many other styles. As a very young man, Zeami was seen performing by the third Ashikaga shogun, Yoshimitsu, who was captivated by the young Zeami and brought him into his household, where he was tutored in a variety of literary and cultural arts. Among the many plays composed by Zeami, some of the most beautiful and well-known are based on the Tale of the Heike, including Tadanori, Yorimasa, Sanemori, and Atsumori. In most of Zeami's plays based on the Tale of the Heike, the protagonist is the ghost of a warrior killed in battle. Summaries of some of the warrior plays and images and illustrations of their performance can be found here: http://www.the-noh.com/index.html. Both noh and Heike biwa enjoyed shogunal patronage during both the medieval and early modern periods, and are considered to be official arts of the warrior class.
Both the Tale of the Heike itself and the various plays it spawned in kōwaka-mai and noh became the basis for reinterpretation of the base stories in the early modern puppet and kabuki theater, where the base stories are wildly reconfigured into sometimes parodic, but always melodramatic, renditions involving numerous new characters, plot twists, and surprises. A summary of a kabuki play based on the famous Battle of Ichi-no-tani (Book 9) can be found here: https://www.kabuki21.com/itfg.php.