Leather dye is not easy to remove once it has been applied. Unlike paint or other topical colorings, it is a stain and is designed to permanently soak slightly into the leather. There are, however, some techniques that can help remove at least some of the stain caused by unwanted dye on leather.
Use a razor, fine-grit sandpaper or a stiff wire brush to scrape the top layer of the leather. Not only will this remove any protective top coatings that may have been applied to the leather after the dye, it will physically remove the top portion of leather, and along with it much of the dye. Doing this will, however, roughen the surface of the leather, and should be avoided on excessively thin pieces so as to avoid scraping a hole in the material.
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Dampen a cloth with denatured alcohol and rub it on the dyed leather. This will remove much of the stain, but only if the dye originally used was alcohol-based. Do not to soak the leather with the alcohol, as doing so will damage it. Use as little as possible to remove the stain, then rinse it in water and apply neatsfoot oil or another leather conditioner to the material to restore the natural oils removed by the alcohol. Because leather is a natural material, you will probably not be able to get all of the stain out with this method and may be left with a splotchy result.
Wipe down the leather with lacquer thinner in the same manner as the denatured alcohol if the alcohol proves ineffective at removing the stain. Lacquer thinner will work on different dyes than will denatured alcohol, but carries the same set of drawbacks: the dye may not ever fully come out, and the leather will be dried out and damaged to some extent as the lacquer thinner removes its natural oil content.
Soak the material in a very weak solution of bleach and water. This will get rid of the dye stain, but will also turn the material itself a pale gray. This method should be saved for extreme cases, as the bleach will do significant damage to the leather itself. After soaking the leather, rinse it well in water and apply several coats of leather conditioner to help restore some of the natural oils.
Dye over the entire piece of leather with a darker color if none of the other techniques prove effective at removing the unwanted dye to your satisfaction. While this will not remove the stain, it will effectively mask it, which is the only available option if solvents and physical removal of the top layer of leather do not work.