In the play, Turner, 56, portrays Sister Jamison Connelly, a recovering alcoholic nun who works as a rehab drug counselor who tries to help a gay teenage prostitute.
The drama will close on Sunday after eight regular performances and 28 previews at the Booth Theatre. "High" earned only $91,020 last week, which was seventeen percent of its potential gross. The show also received some unenthusiastic reviews.
The play is the first open-run show to debut and close this year, but there are currently four shows on Broadway playing to theaters with less than 50 percent of the seats filled.
Robin Williams currently stars in a production of "Bengal Tiger at the Baghdad Zoo, which earned $477,410 last week, which was 47 percent of its potential income.
Turner is best known for her roles in "Body Heat," "Peggy Sue Got Married" and "Romancing the Stone." In 2009, the actress appeared in several episodes of David Duchovny's Showtime comedy "Californication."
Turner has dealt with alcohol abuse in her own life, admitting that she abused it to deal with the pain of acute rheumatoid arthritis.
The actress plays a Catholic mom in the new independent film "The Perfect Family," which premiered at the Tribeca Film Festival in New York last week.