I am a Sri Lankan citizen travelling to British Virgin Islands (BVI). I have BVI work permit. I will be traveling through Doha, Amsterdam, and Sint Maarten.
Do I need a transit visa for either or both of this connections?
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I think you need an airport transit visa in Amsterdam. The general rules are detailed in Do I need a visa to transit (or layover) in the Schengen area? but there are several details that make your situation a bit peculiar.
The flight to Sint-Marteen, which is part of the kingdom of the Netherlands but not of the Schengen area, should be treated as an international flight and that would usually allow you to transit airside. For Sri Lankan citizens, this generally means getting an airport transit visa. But since the British Virgin Islands are not part of the EU despite being a British territory, I don't think that your residence permit would be enough to qualify for an exemption, which is why you do need this visa. It's all pretty exotic so I think it would be advisable to seek advice from the airline or the Dutch consulate if possible.
In Sint-Marteen, no visa is required for transit under 48 hours.