Entering the DR without a passport has never been a problem, but sometimes leaving it without one can be for those who are naturalized American citizens.
This point was brought home to my wife & I during a trip to the DR some 10 yrs ago. In the rush to meet our DR flight at MIA, we accidentally left our passports on the belt at the security x-ray machine. When we arrived at SDQ, no problem being admitted because we both had our Virginia drivers licenses with us.
The problem came at departure. AA would not give my wife a boarding pass merely on the strength of her drivers license. Although she was born & raised in the DR, she had been a naturalized US citizen for some 10 yrs before this incident. But AA refused to issue her a boarding pass without seeing a US passport or a Dominican passport with visa, or at least a US green card. The problem, it seems, is that she looks "too Dominican" (whatever that means). If AA issued the pass and then INS refused her admittance at MIA, she would be put on a flight back to SDQ and AA would be fined. I looked "American" (whatever that means) so the AA agent was "willing to take the risk" with me. [Note: even so, the INS guy at MIA did in fact give me a hard time about not having my passport. He finally waved me through, but not before making me sweat.]
We argued and argued with the AA agent, to no avail. My wife had to remain in Santo Domingo, go to the US embassy and report her passport as stolen, and on the strength of her naturalization papers I faxed to the embassy, she was issued a new passport and allowed to leave the DR.
I know this doesn't happen often, but it only had to happen to my wife once to convince her to never, ever leave the US without her passport again, no matter what local official policy says.
I also have had a couple of times when my US passport saved me from difficulty in places like the DR where I did not need to carry one, but was glad I did.
So I recommend that you take it with you, even if you don't use it to get into the DR. And even more prudent, carry with you (not in the same place as your passport) a photocopy of the passport. When you get to your lodging in the DR, put the passport in a safe, but carry around the photocopy in your wallet or purse. It can come in handy, believe me.
Best Regards,
Keith