"We can't tax landlords, they'll make the tenants pay for it" is not how markets work. Pricing is held down by ability to pay, because - as far as landlords are concerned - getting paid some money is better than getting none.
Longer term, the impact on rental costs depends on whether you think this policy would increase the amount of housing available for people to live in.
Where I live, many properties are bought and kept empty - the rental income isn't worth the hassle and they are being held for capital gains. High property taxes would presumably result in some of those properties being sold to occupiers.
Consider an extreme case - a land value tax of 20% (meaning you pay the government the full value of the unimproved land every 5 years). Land values would plummet, right? Lets say it were introduced over 30 years, so prices could adjust smoothly instead of crashing. What does the steady state look like after 30 years?
