It's NOT possible, if your talking about a driving license as you need a cedula to get a Dominican license. Other than that, you should be fine.
If your coming here on a regular basis, then you will be able to drive on your overseas license. Your license is valid for as long as your tourist card.
You will get many conflicting stories on what you can and cannot do, how long you can drive for etc etc.
According to immigration your license is valid for as long as your tourist visa, 15 days. You can take the risk and drive longer than that, but be warned, your insurance company might look for every loop hole possible if you have an accident. The first thing the cops will do if you have an accident is look at and photocopy the entry stamp in your passport, this is usually attached to the accident report for the insurance company.