Possibly. But so does underused real estate, by reducing supply. And housing is a fairly inelastic good, so even a 5-10% increase in supply could have a significant impact.

Not necessarily. Rent is decided by supply, and demand. Property taxes may encourage smaller homes, which increases supply (as you can build more of them); and demand won't change at all (since people will still spend 1/3rd of their income on rent, or more if they live in NY).

They can just focus on really high tax rates for homes that aren't primary residences.