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The volume of the displaced water would be less - as you're reducing the mass of the boat.

Another viewpoint: I think there's a bit more to this question, but the basic answer remains the same. I think it's all about "Archimedes' Principle".

Let's consider the anchor as still part of the boat.

Also let's think about it before it gets partly buried in the ground underwater.

Archimedes' Principle tells us:

The "upthrust" on the boat before the anchor is lowered is equal to the weight of the whole boat. That equals the weight of water displaced.

The anchor itself doesn't float in water. It is denser than water.

When the anchor is completely submerged it displaces an amount of water equal to the anchor's volume (not the anchor's weight).

When the anchor was on the boat it displaced an amount of water equal to its weight.

So, when the anchor is lowered, the boat (including anchor) displaces a slightly smaller volume of water.

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