The Virgin Islands are two countries, US Vigin Islands and the British Virgin Islands.
The US islands have the same entry requirements as the USA
The British Virgin Islands require visas from people from;
Afghanistan, Albania, Algeria, Angola, Armenia, Azerbaijan, Bahrain, Belarus, Benin, Bhutan, Bosnia-Herzegovina, Burkina, Burundi, Burma, Cambodia, Cameroon, Cape Verdi, Central African Republic, Chad, China (except Hong Kong SAR), Columbia, Comoros, Congo (Democratic Republic), Cuba, Djibouti, Dominican Republic, Egypt, Equatorial Guinea, Eritrea, Ethiopia, Gabon, Georgia, Guinea, Guinea-Bissau, Guyana, Haiti, Indonesia, Iran, Iraq, Israel, Ivory Coast, Jamaica, Jordan, Kazakhstan, Kyrgyzstan, Korea (North), Kuwait, Laos, Lebanon, Liberia, Libya, Macedonia, Madagascar, Mali, Mauritania, Moldova, Mongolia Morocco, Mozambique, Myanmar, Nepal, Niger, Nigeria, Oman, Pakistan, Occupied Palestinian Territories, Peru, Philippines, Qatar, Russian Republic, Rwanda, Sao Tome & Principle, Saudi Arabia, Senegal, Serbia, Slovakia, Somalia, Sudan, Suriname, Syria, Tajikistan, Thailand, Togo, Turkey, Turkish Republic of Northern Cyrus, Turkmenistan, Ukraine, United Arab Emirates, Uzbekistan, Vietnam, Yemen, Yugoslavia and Zaire.
Everyone else can stay for up to 30 days without a visa.
British Virgin Islands visa information