Landlords charge what the market will bear and generally have little if any price elasticity of supply, so they bear nearly 100% of the economic incidence of the tax. This is especially true where housing supply is constrained by zoning policy rather than the price of expanding supply.
Landlords charge what the market will bear and economical for them to own the rental.
I don't think anyone I know will own a business that is not profitable.
Landlords will absolutely own things that are not cash-flowing. Go look around for empty lots in Manhattan.
It'll make it so new construction tends to be condos rather than apartments, since the first is heavily preferred by tax policy. This reduction in supply will mean higher rents.
New construction is like 1-2% of total housing stock annually. It'll take a relatively long period of time for these sorts of changes to hit prices because again, low price elasticity of supply.
edit: on slightly further thought, you'd also need this effect to either increase overall vacancy or reduce total construction, since any supply shift from occupied rental units to owner-occupied condos also implies a demand shift from renting to owning.