Earlier quoted context omitted.
It can't increase rent. Rent are capped by what tenants can pay, they are specifically set to "as much as the tenants can afford" because everyone needs a roof. Taxing the landlord doesn't give more moneys to the tenant. However, what could happen is that the rental income after tax is lower and not enough anymore to cover the mortgage or the maintenance costs, then the property has to be sold.
When that property is sold, from whom will the displaced tenant now rent? Market rents are set by supply and demand, just like most goods. More supply of rental property? Rents go down so that landlords aren't stuck with vacancies. More demand for rental property? Rents go up so landlords can maximize their profits.
FYI: Selling a property doesn't take it off the market. It's just owned by a new owner, who's gonna either live in or rent it just like before.