I am an Indian citizen who came to the U.S. on an F-1 visa to do my masters degree. I am now in OPT (Optional Practical Training) status and looking for a job. Can I go to the Dominican Republic for a trip?
According to Timatic, the database used by airlines with visa requirements:
Visa required, except for Passengers with a valid visa issued by Bulgaria, Canada, Croatia, Cyprus, Ireland (Rep.), Romania, USA, United Kingdom or a Schengen Member State traveling as tourists for a maximum stay of 90 days.
Visa exempt visitors traveling as tourists must buy a Tourist Card on arrival. Fee: USD 10.- for 30 days. Extensions possible for addition fees.
Warning: - Visitors not holding return/onward tickets could be refused
The website of the Embassy of the Dominican Republic confirms that a visa is not required if you "can legally travel or reside in the United States."
Note that this likely won't work if your US student visa has expired. You'll need your Indian passport, a round-trip or onward ticket out of the Dominican Republic, and will have to purchase a tourist card for $10, either at the airport when you arrive or online. This will allow you to stay up to 30 days.
Note that the above is specific to entering the Dominican Republic. Returning to the US is complicated, and it's extremely important to ensure you get it right. Here's some relevant advice from UC Berkeley (starting with a recommendation you minimize unnecessary international travel). Since you're unemployed, it is very important to note that there are limits on how long you can stay in status without a job under OPT, and the linked advice specifically recommends you carry proof of employment, which you don't have. In addition, the time you spend out of the US will count as part of your allowed days of unemployment under OPT.
In short, there are some fairly significant issues here, and while you can go to the Dominican Republic, it is important to ensure you can come back to the US too. I'd give serious consideration to, at least, waiting until you have a job. Consult the international students' office at your university for any relevant information they may have to offer, and consider the potential risks of this trip.