Factual error: After Zidler tells Satine that she is dying, he is seen walking up a stairway past some sewing women. In an overview of the stairs, a stage light to the left has about a dozen or so frosted light bulbs in it. Frosted, tungsten light bulbs were not invented until the mid-to-late 1920s. Any electric light in 1900 would have had clear, carbon-filament bulbs. And no, those are not glass shades over the bulbs.
Factual error: According to the "Eyewitness Travel Guides: Paris" of 1993, the Moulin Rouge only started to be a dance saloon in the year of 1900. When Ewan McGregor arrives in Paris in 1899 the dance saloon is open.
Answer: If Satine runs, the Duke takes control of the Moulin Rouge and shuts the place down, putting everyone she works with onto the streets. If she stays with the Duke, she'll still die, but the Duke won't have any real reason to shut things down. She could run away with Christian and have a few days of freedom and love with him, but she'd be condemning everyone at the Moulin Rouge to destitution, plus the Duke would probably have Christian hunted down and killed. By staying for her final few days, she saves everyone that she loves, at the price of her own happiness (and that of Christian, but at least she'll save his life).
Tailkinker ★