Do I need a passport to go to Tortola?
You haven't said where you're coming from. Generally speaking even if you're a citizen of a country, you need a passport to get there, if you're not there already. In theory you could blag your way in, if you could prove you genuinely were a citizen, but in practice you wouldn't be allowed on the airplane from the country you were coming from, and a passport is needed to prove you are a citizen.
You might be confusing the need for a passport with the need for a visa. But then you'd have to tell us your nationality.
So of course if you're already in teh British Virgin Islands you don't need a passport.
Wikipedia tells me that it's not part of the EU, so non-British European citizens would need a passport. I don't know whether British citizens would need a passport if they came on a direct flight from the UK (and I don't know if there are such flights). Since you didn't say that you're British, I didn't put too much effort into researching it.
Obviously if you're coming from the USA, you'll need a passport or, if going by sea (not flying) a US passport card (United States Passport Card)