Depending on the specifics of the project, a full dry can take up to 72 hours. When paint is applied more thickly, lengthen the drying time. Naturally, a warm environment cuts drying time appreciably.
Placing painted fabric near a fan or air conditioner can speed up this process as well, and many people use hair dryers to speed up drying. Although it's possible to place painted fabric in the sun, it is important to monitor this process to prevent color fading and other unforeseen consequences, such as insects and debris sticking to the paint. Some brands of fabric paint dry more quickly than others.