"The producer is unable to pass the tax onto the consumer and the tax incidence falls on the producer. In this example, the tax is collected from the producer and the producer bears the tax burden. This is known as back shifting."
From section "Inelastic supply, elastic demand" https://en.m.wikipedia.org/wiki/Tax_incidence
If the landlords decided to increase rents expecting tenants would foot the bill, they would find out tenants to leave the location because it would be unlivable for them at that cost.
More likely scenario is by introducing LVT the landlord would be forced to do the opposite - ie decrease rent - as other landlords would dump their uninhabited properties back to the market.