The Crucible has Abby Williams, who remorselessly accuses a woman of witchcraft so that she can marry the woman's husband. It backfires. Spectacularly.
Even before this, it starts to fall into Yandere territory when it's revealed that Abby drank blood in a charm to murder Elizabeth. Although it's subverted in the end, as Abby leaves before John is executed.
While it was mentioned in the literature section, The Phantom of the Opera deserves a mention here as well. The title character is a combination of this trope and The Woobie. Even though he's usually portrayed pretty sympathetically, he still has a number of rather creepy lines.
Erik: Your chains are still mine — you belong to me!
He gains Christine's trust by impersonating an angel sent by her dead father.
Wicked has not one but two scary stalkers. Petrifying clingy girl Nessarose, out to enslave her beloved's people in order to keep him with her; and Boq, the beloved in question, who doesn't understand Glinda's just not interested. She doesn't even remember his name but he lives in hope.
Boq's far more of a Love Martyr, considering that while he acts desperately out of hope that Glinda will notice him, he never actually crosses the line into stalking.
Roderigo in Othello definitely qualifies, chasing Desdemona to Cyprus in his mistaken belief she'll submit. How he expects this to happen when they never actually meet...
The fact that Iago was leading him on probably didn't help to discourage him.
Marius and Eponine both qualify in Les MisÚrables. Eponine stalks Marius, Marius stalks Cosette and uses Eponine as an intermediary in his stalking of Cosette. Only one stalker comes out of it alive.
It helps that Cosette would probably be stalking Marius right back if her Overprotective Dad ever let her out of the house on her own. She certainly obsesses over him at home, and once they finally meet, it's Love at First Sight.
Fosca from Passion is, aside from being a terminally ill Abhorrent Admirer, a Tsundere who emotionally blackmails her target, Giorgio, starving herself when he spurns her and guilting him into writing her a love letter when he finally visits her again. Unusually, Fosca ends up winning Giorgio over - and we're not so sure that's a good thing. What convinces him to go to Fosca is the realization that she would give everything, including her life, for his sake, whereas his current lover Clara won't leave her husband. (This, ignoring that Fosca has one foot in the grave already, and Clara says she'll leave her husband once her infant son is old enough to fend for himself.)
Do they think that walls can hide you? Even now I'm at your window... I am in the dark beside you buried sweetly in your yellow hair I feel you, Johanna And one day, I'll steal you 'Til I'm with you then I'm with you there... Sweetly buried in your yellow hair
Although Sweeney and Turpin's versions of "Johanna" take the creepiness to the nth power. Besides, "stalker" implies that Johanna wanted him to leave her alone. She was perfectly happy to have his attention, if only because she really, really didn't want to marry Turpin.
Yep to above. It's mainly in the film version that Anthony's song takes on a "stalkerish" feel, particularly with his somewhat creepy bloody face as he sings (since he just got the crap beaten out of him by the Beadle). In the stage version, the two of them actually meet for the first half of the song before being interrupted by Turpin and the Beadle (which involves the Beadle snapping the neck of the poor little bird that was Anthony's gift to her). The second half of the stage version isn't nearly as stalkerish as the film version, and has Anthony ripping apart the bird cage in a rather symbolic gesture.
And besides, any stalkerish tendencies that Anthony has towards Johanna are pretty much kicked to the curb in the musical when you look at Judge Turpin, given the uber-Squick of "Mea Culpa". To accuse Anthony of being a perv after that is like being violently attacked by an anaconda and then freaking out at the sight of a grass snake.
Also, Mrs Lovett. Not so overt, but definitely there. She's clearly obsessed with Sweeney, and has been for a very long time. Complete with cooing over how "beautiful" he was fifteen years ago, keeping his straight razors in case he ever came back and needed them, and (in the film version at least) keeping his old room exactly the way it was when he left it. Yikes.
Don José in Carmen after she gets tired of him. He eventually kills her when she refuses to take him back.
Der Tod (Death) in Elisabeth. He spends most of Elisabeth's life stalking her because he's very attracted to her.