The Dominican Republic left the Inter-American Court of Human Rights because of a decision about granting citizenship to the children of Haitian immigrants. Leiv Marsteintredet has an extensive discussion about the issue.
This is a dark moment for the Dominican democracy and the protection of the most vulnerable groups living in the country, the Haitian migrants and Dominican-Haitians. It is also a dark moment for the IACtHR and the whole IASHR, which now results weakened. Even thought the Dominican Republic may be a small and unimportant country in Latin America it is still the most democratic and democratically stable country to ever leave the IACtHR. Peru left in 2000 under Fujimori, an act that never took effect since Fujimori resigned not long after, and Venezuela withdrew under Chávez (in addition Trinidad and Tobago withdrew in 1998). The Dominican case, I fear, is likely to be the worst blow of them all. Peru returned after re-democratisation with Fujimori's resignation, and it is not unlikely that Venezuela would return should the Maduro-regime fall or the opposition win a future election. Today, however, the Dominican Republic is led by its most international human rights friendly government in decades, if not ever, and the regime is a stable electoral democracy, not any form of populist authoritarian regime. It therefore seems very unlikely today that the Dominican Republic will return any time soon.
Aside from the particulars of the case, we're now seeing a country leave the IACHR not because of ideology (such as Venezuela, which claims conspiracies) but simply because it didn't like the outcome. This follows Colombia's decision in 2012 to pull out of the International Court of Justice because it did not like the ICJ ruling on Colombia's maritime border with Nicaragua.
One would think that in this more democratic era, Latin American governments would be more willing to accept decisions made by international institutions. Yet they're going beyond mere disagreement. I think that this merits collective attention in the hemisphere--Venezuela will rant about U.S. control, of course, but all governments need to discuss why they're suddenly ditching these commitments.
Adhering to international institutions requires giving up some sovereignty. Unfortunately, the United States is in a particularly bad position to lecture anyone on that topic, but should be part of a discussion about how a regional human rights regime should function.
Boz has a quick take here as well.