In the US, the home mortgage interest deduction does not apply to rental properties. To qualify for the deduction, you either have to use the home as your residence for a minimum period each year, or rent only part of it (and meet requirements for renting part of the home: you can't have more than two tenants, and the part of the home they live in can't have its own separate kitchen or toilet facilities).

But that's the home deduction. Apartment buildings, for example, are investment property for which mortgage interest is a deductible expense. https://www.irs.gov/publications/p527/ch01.html

Then you concede that taking away the home mortgage interest deduction (making the homeowner-who-resides-in-the-owned-home and the renter equal for tax purposes) would not drive up rents, and your original comment was at worst wrong and at best a non sequitur.

I'm positing false equivalence. Taking away the home mortgage interest deduction would make the homeowner and renter unequal. Doing so would not drive up rents, just as eliminating the mortgage deduction on investment property wouldn't impact house prices.

You said:

How would a renter feel if their rent increased because their landlord was unable to take a mortgage deduction?

I countered that the home mortgage interest deduction doesn't apply to rental properties. Renters and homeowners are thus currently unequal from a tax standpoint, since renters effectively pay a higher tax rate than homeowners. Removing the deduction would thus not create a new inequality; it would remedy an existing one.

The home mortgage interest deduction specifically doesn't, but mortgage interest on rental properties is still generally deductible.

https://www.irs.gov/publications/p527/ch01.html