If it's s limited resource which people require (shelters), the price (rent) will be quite elastic. There will still be competition if the taxes go up. Competition is what prevents the price from rising unnecessary. The floor for all the competitors went up because taxes went up and they all want to stay profitable. They aren't going to take loss, so they raise rent to cover it. Buying won't look very attractive either because the tax increase raises those costs, and if it's as significant as they other poster wanted (to affect rich people) then it will likely exceed the $10k SALT deduction limit.

This SALT deduction is part of how they penalize landlords already, which is just passed on in the rent payments. This is part of the reason renting is more expensive than buying (comparing just monthly payments), in addition to management fees, profit, and repair/renovation reserves.