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.

If they're losing money at market rents, what are they going to do about it? If they could raise prices, they would've raised prices before the tax because that's just leaving money on the table. If they throw an emotional fit and leave the unit empty, they're losing even more money. If they sell or go bankrupt, the new owner will have bought it at a lower price and so end up having a profitable business at market rents.

If renting out those units can't be profitable, they'll get sold to a market where people own them instead, not rentals.

Nothing particularly wrong with that, but it does reduce the amount of rental units in town, increasing the number of owner-occupied units in town.

Some people do prefer to rent so they might be unhappy about that. Some people prefer to own, so they'd be happy about that.

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.