Are you planning to paint the whole dresser and if so what kind of paint are you using? If you are using an oil base enamel,just clean the piece with a good all purpose cleaner like Murphy's Oil Soap.Sand any rough spots and paint.If you are using acrylic or latex paint(which I prefer)you will need to clean the piece with a product such as trisodium phosphate or some other cleaner that will remove all traces of the oil.Acrylic and latex will not adhere to oil.If using trisodium phosphate follow directions on the container carefully and wear protective clothing.Sand any rough areas and apply a sealer like Kilz.A hint about painting.Use a small sponge roller rather than a brush.It will leave a smooth finish with no brush marks and painting will go faster.You can use a brush for hard to get to areas.........Lillie

Thanks for your help, Lillie. I haven't had much experience in this field, but it sounds like going with an oil based acrylic might work better in this case. I'm going to use your suggestion about using a roller as well as I hate brush marks.
I think I'll start with the side that won't show, just to see how it comes out :-)
Lois

Oops, I meant an oil based enamel, not acrylic. I'm getting my paints mixed up.

Do not scrub with Murphy's - it's OILY and the paint will not stick.

Scrup it with TSP to get the oil off, then prime and paint.

Sounds like it's going to be pretty. Let us know how it turns out.

TSP is trisodium phosphate,right? I knew there was another name for it.LOL Other all purpose grease cutting cleaners will work also,like "409",the orange cleaners,etc. and are easier and safer to use than the TSP.You could also use a steel wool with the cleaner to help remove any oily buildups.
Can you put the crackle finish on oil based enamel? I'm not sure on that one so you might want to check that out.I know the "crackle" step is water base.
I'm anxious to see the finished product so be sure and post a picture when you are done............Lillie

I did find a method online that allows the use of oil-based paint to do a crackle-type finish. It involves a layer of oil paint as a base coat, followed by a layer of water-based white glue, like Elmers. Then you go over it with a layer of milk-paint, followed by an oil-based glaze after it's dry.
I like this idea as milk paint colors are usually nice historic looking colors, and there won't be as many to choose from so it will make the choice easier. That's where I have usually have the most trouble, and it will be easier to choose from 20 colors than from 1000 :-)

I'm going to try a sample board first to make sure it will look the way I want it to. I have a piece of oiled oak in the basement that came from my son's old desk, I think it was a shelf that never got used. I'll treat that to the kind of cleaning you all recommended, and then test the painting technique on it.

Now to choose a color and get started! Thanks all!

Here is a link that might be useful: Antique Paint Finish

Is there a glaze I can use over an oil based paint?