How do you paint furniture then remove the paint to make it look old?

You could paint it with a color of your choice, then you could use crackle paint on top in an antique white color, or use another color on top with broken brush strokes and finish with crackle glaze that allows the color(s) underneath to show where the glaze cracks.

Here are some ideas.

For a rustic driftwood effect use white vinegar and a steel wool pad

There are a number of methods that you can use to create an aged look. I like to use Annie Sloan Chalk Paint and then lightly distress the paint with sandpaper on the areas that would naturally be worn over time. The paint needs to be sealed with wax. You can also apply a glaze or dark wax to further age the finish. If you like a heavier chippy look then Milk Paint is an option. I share the techniques I use to create various finishes, so if you want then head over to my blog: http://theblacksheepshoppe.blogspot.com/