Can a person in the USA in possession of Green Card and US Travel Document (I-571) travel to British Virgin Islands for tourism for a week without having to go through the British Consulate in the USA / visa procedure? Person travelling has an expired Russian Passport which cannot be renewed in time for travel.
Although the USA Today is not a government institution, it seems the official page corroborates the information: you don't need a visa but you do need a valid passport.
Green card holders visiting the British Virgin Islands are granted entry without visas as long as travelers meet the standard travel requirements. Visitors are allowed into the islands for a month provided the correct documents are shown at a land or sea port of entry: a passport, proof of accommodations on the island such as a hotel reservation and proof of your return journey such as an airline ticket. If you're a green card holder, you'll need to use a passport from your country of origin.
BVI Tourism somewhat corroborates although doesn't mention a green card:
European citizens need a valid passport to enter the British Virgin Islands.