It's likely to be hot and incredibly humid in June which, on average, has more days with significant rainfall than any other month in the year:
http://www.bbc.co.uk/weather/world/city_guides /results.shtml?tt=TT003190
The official hurricane season also runs from June to November. Although the highest risk is around August and September, there's still the possibility of a hurricane striking in late June.
If you really want to go to the Dominican Republic, I'd recommending waiting until you're able to travel between December and April.
Chris