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.

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.