Property owners generally charge as much rent as they can. Only in places where it is easy to build new apartments is there floor on rents. In that case the floor is a combination of constructions costs, interest rates, and the rate of return to investors of other opportunities in the area. So in a place where building is hard, the rents would probably not rise, but in a place with lots of land and easy zoning, the rents probably would.

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.

From the new landlord who bought the place, supposing they didn't buy to occupy it themselves.

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.

Of course it would, either directly or via a reduced supply of profitable rental properties (and the resultant shift in supply-demand equilibrium).

There may be an offsetting overall reduction in the value of all real property, but it seems like reducing or eliminating these deductions[0] would be harmful to renters, not helpful.

[0] - Deductions like this, by the way, are available to all sorts of other profit seeking businesses for the equipment and supplies they use in the conduct of their business and I see no reason why a house should be different from a factory machine or laptop computer or airliner nor a minor repair to a house be different from a pad of paper or other consumable with regards to whether ordinary and necessary business expenses ought to be deductible against gross revenue when computing profit.