I'm not sure that the DR has any area that would qualify under a common scientific definition of rainforest -- certainly nothing equivalent to the rainforests of Brazil or Costa Rica. However, there are a few areas referred to as "subtropical rainforests," such as part of Los Haitises, and I know some tour outfits in the DR promote this as simply "rainforest," but I think this is misleading advertising.
The DR does have an unusual, interesting and beautiful "cloud forest" in the Loma Quita Espuela scientific reserve near San Francisco de Macoris, considered to be the "mother of all rivers" in the DR. Worth visiting if you're a hiker.
Hope this helps, Alan.
Best Regards,
Keith
P.S. BTW, in the sticky entitled "Some Useful DR-related Environment Links" at the top of this Forum, there are good links to sites that tell you all about the DR's national parks and reserves.