Typically inelastic goods mean that prices can increase because demand will not drop as prices go up.
There are two markets with two types of consumer here:
1) The market for a roof over your head. This is highly (but not completely) inelastic. People need somewhere to live.
2) The market for property as an investment. Higher taxes on an asset disincentivizes ownership of that asset. This is elastic because there are tons of other asset classes you can pour surplus wealth into.
Effectively a land value tax already exists for renters - in demand locations have higher rents. The "tax receipts" just dont flow into government coffers.
Depending on how high and fast it was raised it would likely lead to a steady stream of landlords defaulting on their mortgages and leaving the banks holding the property.
This is actually very easy to see in countries with no 30 year mortgages, because landlords have to refix their mortgage rates every 1-5 years. This means you can see regular landlord cost changes as interest rates fluctuate, as well as the impact on rents.
The impact on rents is essentially 0 as landlord costs change; instead, rents follow tenant incomes, because landlords are able to charge more as long as tenant incomes increase.
I've got plenty of data from New Zealand if you're interested, but I'm sure it's the same everywhere there is supply-constrained housing.
How do you explain rising rents?
Tenants have several fixed costs to pay: taxes, food, rent, and transportation being the main ones. If their incomes go up, rent typically consumes the increase, because housing is a fairly uncompetitive market.
As much as the market permits, as the rental market also competes with actually just buying the house.