Renting the apartment in SF will not be cheap because the land has extremely high value. It's just the government will capture the high value of that land instead of landlords and that high value land capture will be used to remove other taxes like income tax and property tax. Land prices in SF would fall slightly as the LVT removes most speculation. So likely rents go from $6000/month to something like $5400/month. But the government might capture $4500 of that to reduce other taxes.

If we still had 1870's small government the idea would be to put most of the LVT into a Citizen's Dividend which functions like a UBI and helps people pay for their cost of living (including some of the land taxes).

Rental prices would not change, it's just that more of the money would go to government instead of the landlord, especially for low density housing.

They probably would decrease somewhat, because there's no longer an incentive for landowners to prevent people from building housing to keep the supply low.

This is something I’ve been wondering about with LVT. Wouldn’t they still have an incentive to stop people from building more dense housing near them? Even more of an incentive than now, in fact? Because the more people that can live and work in the immediate area near them, the more valuable the land is, and the more they’ll pay in tax.

As the land becomes more valuable, it becomes more valuable to them---better access to services, etc.

If it becomes more valuable, but in such a way that they can't leverage it, they can sell it.