Depends on where the island is?

Where is said island?

What country lays claim to the island?

Larry Ellison (owner of Oracle) bought an island in Hawaii (Lanai) in 2012 – he paid something like $300 million.

Since this island was in the borders of that state, he paid some massive amount in taxes on the sale and on registering his ownership.

If he had decided to find an island that no country has claims on, he might not have to pay taxes. But that would be difficult.

Most islands would be part of a governmental jurisdiction and you pay taxes to that government.