Spacey, who stars in the political drama, has been accused of making sexual advances to a 14-year-old boy.
The show was already due to end after this season, but production is now suspended "until further notice".
The Old Vic theatre in London, where Spacey worked for 11 years, said it was "deeply dismayed" by the allegations.
The 200-year-old theatre has set up a confidential complaints process for anyone connected to the Old Vic to come forward.
