The Joker feels this way about Batman, according to Frank Miller. All of his crimes are part of his "game" to get the attention of The Dark Knight, who he calls "my sweet".
In Death of the Family Catwoman calls him out on it. At least in Scott Snyder's part of the storyline, she doesn't even really have to. Not only does the Joker refer to Batman with romantic epithets, and not only is Batman unnerved when he sees affection in the Joker's otherwise emotionless eyes, but the Joker's campaign against the Bat-family has 'definite' romantic overtones, such as forcing men in Batman and Joker outfits coupled together to dance for days.
Bruce's childhood friend turned villain Thomas Elliot who became Hush. He seems to obsessively stalk Bruce and got plastic surgery to look like Bruce.
Two-Face gets a sort of sick obsession with Renee Montoya after their experience in Batman: No Man's Land. When he learns that she's actually gay in Gotham Central, he just makes it another part of his plan to destroy her life so she has nothing left but him.
Post-Flashpoint, Mr. Freeze is reimagined as one of these. Victor Fries was hired by Wayne Industries to study a woman named Nora Fields who was one of the first people to ever be cryogenically frozen. Victor developed a twisted interest in her, deluding himself into thinking they were husband and wife. Bruce Wayne eventually shut down the project because he was becoming increasingly disturbed by Victor's unprofessional interest in Nora. Victor performed his own private illegal experiments anyway. When Bruce confronted him again, Victor lashed out causing the accident that turned him into Mr. Freeze. When Batman and Freeze face off in the present, Batman calls out Freeze on his delusional "love" for a woman he doesn't even know who is technically old enough to be his grandmother.
Abraham Pointe in Brightest Day was obsessed with a casino waitress named Lisa and stalked her afar with a camera for two years, mistaking it for love. This could have ended very badly for Lisa. Fortunately (well, sort of), the Predator, the Entity of Love, mistook Abraham's insane crush for love as well and possessed him, which got the Star Sapphires and the Green Lanterns involved. Eventually Carol Ferris called out Abraham on his stalker behavior and said to his face that he never really loved Lisa — he didn't know anything about her except for her first name. Abraham had a Heel Realization after understanding that he didn't really love Lisa, which gave Carol an opportunity to separate him from the Predator. You know a (non-existent) romance is messed up when an Eldritch Abomination getting involved turns out to be a good thing.
In Convergence, Flashpoint Aquaman is hunting after the pre-Flashpoint Mera to replace his Mera, who was beheaded by the Flashpoint Wonder Woman.
Green Arrow: Cupid aka Carrie Cutter for Oliver Queen. While not the sanest person in general (she killed her hairdresser after getting a perfect haircut so no-one else could ever have hair that good), she takes the crazy to new heights thanks to her obsession with Ollie. In an effort to win his love, she has been killing off the members of his Rogues Gallery. She's done pretty well too; she even killed the superpowered crime boss Brick with a wrecking ball. She also tried to get Ollie to kill his Evil Counterpart Merlyn, thinking that this would make him love her. When that failed due to Ollie following Thou Shalt Not Kill, she chained him to subway tracks and planned to die with him. While Green Arrow and Black Canary were able to get out of this situation in one piece (Merlyn wasn't so lucky — Cupid slit his throat as a distraction), Cupid hasn't given up yet.
Dick Grayson aka Nightwing who attracted the attention of Tarantula an Anti-Hero who became obsessed with him to point where she manipulated an encounter with Nightwing in such a manner that Barbara Gordon, his girlfriend at the time, broke up with him. Even getting arrested didn't keep Tarantula away from Dick as she broke out of jail and continuously got involved with his crime-fighting pressuring him to break the no-killing code and when he poisoned and left immobile on a roof top Tarantula took the advantage by taking his pants off, sitting on his crotch and infamously raped him. It's especially jarring as prior to this Dick Grayson being a ridiculous Chick Magnet was Played for Laughs, but this event made Dick feel disgusted and ashamed.
In Nightwing (Rebirth), Raptor is this for Dick's mother, Mary. She stole medicine to cure his leprosy and he fell head-over-heels for her. It's implied they had a relationship at one time but she broke it off. Her murder hit Raptor hard and it shaped his persona. He then took up stalking Dick to "protect" him and carry out the "justice" he feels Mary would have wanted.
Robin (1993): Tim Drake attracts the attention of Mafia Princess Darla Aquista who eventually forced a kiss on him at school. After her death and resurrection she used her new abilities to destroy cars and threaten old classmates to find out Tim's new address then showed up on his doorstep and tried to talk him into helping her kill Robin (not knowing that Tim is Robin) in order for them to start their "new life together" even though he repeatedly states he has no interest in her. She doesn't even think committing homicide in front of him will get in the way of her plans since she "know[s he'] upset but [he]'ll forgive [her] in time". Tim has to call in super-powered help to deal with her.
Powerboy became obsessed with Supergirl in "Identity/Breaking the Chain" after catching a glimpse of her as a Female Fury in The Supergirl from Krypton (2004). He created a shrine and became a superhero to impress and seduce her. However, he grew more and more possessive, and beat the crap out of her when she tried to visit Captain Boomerang (Owen Mercer) in the hospital. Supergirl defeated him in spectacular fashion.
Superman has been stalked by various women, including Maxima, The Queen Of Fables, and Obsession.
Heck, Lois Lane herself was often portrayed as this, intentionally or not, during the Golden Age and Silver Age (and even a little bit into the Bronze Age). Modern writers have struggled ever since to push her away from that.
Pre-Crisis Lana Lang was often portrayed as this, too. She was infatuated with Clark Kent since he was Superboy, and she never stopped. Post-Crisis writers usually portrayed her as someone who had a crush on Clark long ago but got over it and is okay with being his friend.
Superman/Supergirl: Maelstrom: The titular villainess is obsessed with becoming Darkseid's bride. Darkseid punishing her for daring to speak to him, pointing out she is unhealthily obsessed, or bluntly saying he is not interested in anybody do nothing to dissuade her.
Darkseid:Your obsession is unhealthy, Maelstrom. Maelstrom:Such is my love of the great Darkseid. Darkseid:I have no interest in taking a concubine. I've learned that women cannot be trusted. Besides, I doubt you'd survive the initiation process. Maelstrom:I would consider a death such as that to be a great honor.
Teen Titans. The second Clock King is obsessed with Rose Wilson aka Ravager. He murders his lover, The Disrupter, for trying to kill her.
Wonder Woman (1987): While Artemis was the current Wonder Woman due to the events of The Contest and Diana was working with PI Micah Rains she responded to a client who was worried her stalker was going to kill her or her fiancé, and who had already killed her pets. When Diana arrived to talk to the client the stalker was hiding out nearby in some bushes and tried to argue with Diana that he loved his victim which Diana refuted.
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