This is a subject that comes up quite frequently because children seem to discover these pesky little drawing items and color our beloved dolls. Or it could be a case of going to a thrift store and finding a doll that has been marked on. However it got there, from drawing or from dye transfer from clothing it becomes inevitable that we want those marks off. Number one because marked dolls have significantly less resale value if at all, and number two it's an eyesore on your doll.
First we always suggest when starting to clean a doll that you use plain water first. This will remove any surface dirt, grime, and spots without chemicals and expose areas that you may need to treat with more complex methods and chemicals. You want to use as little chemicals as possible, and only when absolutely necessary to preserve the integrity of your doll.
The most popular method and probably the most time consuming is the ever popular Zit Cream method. Yes folks, remedies for getting rid of acne is the same formula that gets rid of dye and marker, somehow lifting the stains. In order to make this method work, you need time and patience and a lot of sunlight. Yes indeed this method needs the heat and light to get these stains out. Don't ask me the science behind it, but apply Zit Cream to the affected area and bake in the sun. Repeat as necessary. Sometimes it takes more than one go to get that stain lifted and off. Twin Pines Remove Zit is a popular brand for this. Note that you don't want to keep it over 6 months of the expiration date as it will lose it's effectiveness. Key ingredient you are looking for is Benzoyl Peroxide if using other brands. Sun treatments can last anywhere from 4-5 hours in direct sunlight.
Other reported methods:
Hand sanitizer. This is one I heard about recently that people were getting results from using hand sanitizer. Note that hand sanitizer has a large amount of alcohol in it's contents and alcohol does strip vinyl down over time. While I have not tried the effectiveness of this method, I caution people who are trying this because you have a chance of ruining the vinyl. Alcohol will remove the sealant on vinyl and will dry out the vinyl in that spot where you used it causing the vinyl to get brittle over time and cause your doll to age faster. That sealant is important to keeping your doll clean as well and removing that barrier will cause your vinyl to absorb more grease, dirt, and grime. It essence, you just stripped your doll of it's protective layer.
Nail Polish removers reportedly get rid of a lot of marks but you want to use a non-acetone version as acetone does the same thing as alcohol where it will strip the protective sealant from vinyl and cause it to become brittle and absorb more dirt over time.
The thing to continuously check for before using a chemical on your doll is whether it has been tested safe on vinyl and plastics. This will guide you in the right direction to not damage the vinyl on your doll.