The Dominican Republic does no background checks on the millions of tourists that come here. For an American, and a lot of other nationalities, all is needed is a birth certificate and a photo ID to enter. If your friend is foolish enough to contact the Dominican consultant and inform them that he is a convicted felon that wants to visit their country then it is very possible that they will flag him for non-entry as that is their prerogative.
The American State Department doesn't care if you are a felon or not when applying for a passport and as that is the only legal US document you need for entry to the US it is imperative that you have one for reentry. You DO NOT have to use that passport for entry into the DR.
Of course the failure to have an entry stamp on your passport when you get back to the US might raise some flags it is a nonpunishable offense.
Rick