Sand it out and fill it with wood putty if the damage is deep. Then stain & top coat to match.
You'll have to touch up. If stained you'll need to sand off or fill with stainable wood fill, restain and varnish. If painted basically the same except Prime and paint.Before you do those have you tried to wash off with a baking soda scrub? Maybe lucky and it's only on surface. Good luck.
Take the cabinet door to your hardware store and ask for their advice...they may have products or ideas you never thought of and can help find an exact match of color
This post has some great tips for you.