Bird is (mostly) right. I made a basic error.
The combustion chamber is not bored. That is, the numerator of the combustion ratio (displacement + combustion chamber volume) goes up due to the bore, and the denominator doesn't. So, there is a change in static CR, assuming the piston shape is the same. If it isn't, you can make the CR do just about anything you want.
Now, it also assumes the piston at TDC is level with the deck, which doesn't have to be so. If the piston is below the deck at TDC, the combustion chamber IS larger (assuming for simplicity that the piston has a flat top). The limiting case (all the combustion chamber inside the block) is the only one where the CR doesn't change. You could build a cylinder that way, but it wouldn't work very well. Most real cylinders have a piston height very close to zero. Piston height is not changed by boring.
Now, these things are not at all hard to calculate, and anyone contemplating boring to raise compression really should calculate the effect. All you need is the bore, stroke, and stock static compression ratio to estimate its effect. You can do without the CR (which often isn't very accurately spec'd) if a fractional (percent) change is enough.