The Hindi-language film, called Sins, is set for release on Friday.
"Religion needs to be a personal affair and should not be a subject for entertainment or for commercial use," Joseph Dias, general secretary of Catholic Secular Forum, said in a statement.
He said Christians would demonstrate in front of cinemas planning to screen the film and various Catholic groups had begun collecting signatures to support demands for the ban.
Director Vinod Pande said the film was not intended to hurt anybody's sentiments.
"It's about a forbidden love. There was no agenda whatsoever to hurt anyone," he said.