What changes Wiesler's mind about saving Dreyman?

TL;DR: The script does not reveal the inner motives for Wiesler's transformation in the film. One can assume, however, that Gerd Wiesler was always concerned with a dutiful and selfless performance of his service. A selfish act of a civil servant seems to resist him. For this reason, he withholds information after learning that the surveillance of Dreyman is mainly based on private and career-related reasons of his superiors.

But, in order to get closer to his motives, one can look at Bruno Hempf, Anton Grubitz and Gerd Wiesler:

Bruno Hempf He is a influential minister (a precise official designation is not given) and a member of the Central Committee of the SED. He orders the operative procedure against Dreyman because he desires his partner Christa-Maria Sieland and wants to eliminate his rival. When the actress turns away from him, he uses her addiction to pills to put the artist couple under pressure.

Anton Grubitz He is a ambitious and cynical careerists. The intelligent, but unscrupulous MfS lieutenant colonel is responsible for monitoring cultural workers and in charge of department XX/7. He commissions Wiesler to supervise Dreyman, because he hopes for a further promotion in the MfS. They were former fellow student, so he knows that he could always rely on Wiesler's precise performance of duties.

Gerd Wiesler The MfS Captain has been entrusted with the management of the operative procedure against Dreyman. He conscientiously fulfills his mission at first. Increasingly, the spy touches on the lives of those being monitored - both artistic and private. When he learns that his mission is not based on political or state-preserving, but on private purposes, doubts about his mission awaken in him. Therefore he begins to withhold information.

In the following scenes the character change becomes clear:

Scene 10 (00:38-00:45) Dreyman writes. - Wiesler sketches Dreyman's apartment in the attic. - In the car, Hempf harasses Christa-Maria. - Dreyman sees his girlfriend getting out of the minister's car. - The actress collapses crying in the bathroom. - Dreyman plays piano (music and real music). - Christa-Maria swallows tablets. - In bed, Dreyman embraces her in silence. - Wiesler adopts the same posture as Dreyman.

Scene 11 (00:45-00:49) Wiesler washes his face (the doorbell rings). - He uses the services of a prostitute. - Secretly, Wiesler visits Dreyman's apartment (music). - Wiesler returns home. - When Christa-Maria tells him that Hauser's lecture tour to the West has been cancelled, Dreyman is not surprised. - Udo takes notes. - Dreyman is looking for a tape by Bertolt Brecht. - Wiesler reads this book at home (Voice Over = VO: Dreyman, music).

Scene 12 (00:49-00:54) In the morning, Dreyman's friend calls Wallner and reports that Jerska has hanged himself. Shocked, Dreyman plays "The Good Man's Sonata" on the piano. - Wiesler eavesdrops deeply moved (camera moves in a semicircle around him). - In the elevator, a boy asks him whether he is with the Stasi.

Scene 15 (01:04-01:13) At Jerskas' funeral, Dreyman drafts an intellectual text about suicides in the GDR (VO: Dreyman). - He writes it down at home (VO: Dreyman). - Dreyman visits Hauser, who turns the music up loud because of the Stasi (real music). - He gives the text to Hauser and Wallner to read. - With the help of Hauser's uncle, who pretends to smuggle his nephew into the West, they test whether Dreyman's apartment is bugged. - Wiesler wants to report this by telephone, but hesitates - Hauser's uncle calls from West Germany. Dreyman thinks he's safe. - Wiesler falsifies the report.

Scene 16 (01:13-01:21) The "Spiegel" editor Hessenstein discusses the text with Dreyman and Hauser. - Wiesler understands that Hauser is not in the West. He tells Udo that his friends are writing a play. - When Christa-Maria comes in, she realizes that she is not wanted. - Hessenstein hands Dreyman a new typewriter. - Wiesler types the report (music; PM). - Grubitz shows Wiesler a dissertation on prison conditions for artists. - Wiesler withholds the report. He demands a reduction of the operative procedure. Grubitz agrees.

Note: The information about the characters and description of the content was taken from the following brochure (german): Marianne Falck: Das Leben der Anderen. Filmheft der deutschen Bundeszentrale für politische Bildung (bpb - 2006) (Marianne Falck: The lives of others. Film booklet of the German Federal Agency for Civic Education (bpb))