"The Birthmark" has a very sinister tone.
From the beginning, Aylmer's obsession with the birthmark on his wife's face doesn't bode well for any of the characters. He prevents happiness and accord in their marriage because he cannot let go of what he sees as a blemish and imperfection. It's clear at the outset that it won't be a happy story. This sense of something being wrong and things going poorly makes the tone sinister from the beginning.
It's important to remember that the birthmark itself was not horrid and other suitors had actually admired it in the past. Aylmer's focus on and hatred for the mark, however, magnify it in his wife's mind. They make it into something horrible when before it was something that hadn't truly bothered her. The obsession itself is sinister and is the centerpiece of the story.
The tone gets darker as the story moves on. Aylmer dreams that he has to cut his wife down to her heart to remove the birthmark. She realizes that she might have to die to lose it—and accepts it as the price of perfection. Aylmer's experiments going wrong and causing negative effects is another thing that makes the tone dark and sinister; they foreshadow Georgiana's eventual fate.