To extend your tourist card anywhere from 15 days to three months the official way, you can apply in Santo Domingo at the Dirección General de Migración at least two weeks before your original card expires. Alternatively, you can pay a fine (starting at $10) in the airport when you leave the country.
At the Dirección General de Migración you’ll be required to fill out a form – usually available in Spanish only – and to present your passport, a photocopy of your passport’s information page(s) and two passport-size photos of yourself. The fee starts at US$10 depending on how long you will extend for; your passport and new tourist card will be ready for pickup at the same office two weeks later. The process is a good way to blow an entire day.