The term bunny boiler designates a jealous or obsessive person, originally a woman, whose behaviour in pursuit of a former or intended partner is considered desperate or dangerous.

It alludes to a notorious scene in Adrian Lyne’s American film Fatal Attraction (1987), in which the character of Alexandra ‘Alex’ Forrest, played by the American actress Glenn Close (born 1947), boils alive a pet rabbit belonging to the daughter of her erstwhile lover, Dan Gallagher, played by the American actor Michael Douglas (born 1944). (The term bunny boiler does not actually occur in the film.)

The earliest instance of bunny boiler that I have found characterises the character played by Glenn Close in Fatal Attraction; it is from Close encounters, published in the Liverpool Echo (Liverpool, Lancashire, England) of Thursday 9^th March 1989:

In her latest movie role, Glenn Close plays a woman as vengeful and ruthless as her notorious character in Fatal Attraction. Though she doesn’t actually boil anybody’s rabbit.
The Marquise de Merteuil prefers the subtle approach when it comes to getting her own back while indulging in Dangerous Liaisons*. She makes her point with wicked wit and poison pen rather than a carving knife.
But Ms Close does not necessarily agree that there is an obvious similarity between the French aristocrat and Alex the bunny boiler.

(* Dangerous Liaisons is a 1988 American film directed by Stephen Frears, starring Glenn Close, John Malkovich, Michelle Pfeiffer, Swoosie Kurtz, Mildred Natwick, Peter Capaldi, Keanu Reeves and Uma Thurman. It is based on the theatrical adaptation by Christopher Hampton of Les Liaisons dangereuses (1782), an epistolary novel by the French author Pierre Choderlos de Laclos (1741-1803).)

The second-earliest occurrence of bunny boiler that I have found is used in its current generic sense; it is from the column Dating, by Laura Kavesh and Cheryl Lavin, in The Gazette (Montreal, Quebec, Canada) of Sunday 7^th May 1989:

One tantrum doesn’t make you a bunny boiler

Does one temper tantrum a Fatal Attraction make? Just because a woman makes a scene at one little party, does that mean Glenn Close is her role model? Some people think so.

Laura Kavesh and Cheryl Lavin then recount how Amanda, a young woman, reacted when her boyfriend, Donald, broke up with her during a party at his apartment:

She screamed, she cried, she accused him of leading her on. She did a lot of things that she is ashamed of, but — remember the movie? — she did not boil any bunnies, she did not kidnap any children, she did not attack anyone with a carving knife.
Which is why she was so offended when Donald told her she was “weak and dependent and just like that character in Fatal Attraction.” Amanda knew he wasn’t talking about Michael Douglas.
A funny thing happened. When she told her friends what he had said, it seems that quite a few of them had had the same experience. As soon as they showed any emotion, their boyfriends accused them of “pulling a Fatal Attraction.”
“It’s like whenever men can’t understand a woman’s emotional outburst, they attribute it to psychotic behavior,” says Amanda.
When Donald accused her of acting like Glenn Close, Amanda knew she had to get away. She sublet her apartment, quit her job, moved away, and got into therapy.
That doesn’t sound psychotic.
It sounds pretty healthy.

The term bunny boiler characterises an undesirable man in this personal advertisement, published in The News-Leader (Springfield, Missouri, USA) of Tuesday 12^th March 1991:

DWF 23, 98lbs, blue eyes, attractive. Could make fatal attraction part two from some of the men I’ve met. So if your [sic] strange and a bunny boiler, don’t write. 26-38 for friendship, maybe more. Professional, honest, adventurous, maybe you? Box M-2932, c/o The News-Leader, PO Box 798, Spfld, MO 65801.