Almost any new market rate housing, even projects that don't directly displace anybody, will increase housing demand more than they increase supply, so the end result will be upward pressure on rents for everybody. ... And by making residential investment so profitable in gentrifying neighborhoods, new market housing makes the problem worse.
 — comment, via @MetroObserver via Michael Blume

Their argument is something like: when you build housing supply goes up. Demand also typically goes up as well, because you've made the area nicer, which we'll call induced demand. In most places, increasing supply by one housing unit induces some demand but less than one unit's worth. San Francisco is not most places, however, and here building one more unit induces more than one unit of demand. So building more only makes things worse.

If you accept this, then all things being equal you should favor destroying existing units, say when the current tenants move out. The idea is that each unit destroyed would decrease demand by more than one unit, lowering rents. My guess is, however, that the people who make this argument believe that each unit destroyed decreases demand by less than one unit, and so also raises rents and makes the problem worse. This isn't possible: adding a unit and removing a unit can't both result in higher rents.