Put another way, I think the main thing that is going to take hit as the LVT % increases is not renters or landlords, but the land value itself.
A piece of land that rents for $3100/mo may have landlords willing to pay $2000/mo for the land so they can break even after construction, overhead, etc. Absent any taxes, maybe they could pay the whole $2000/mo in interest on a loan to buy the property. As the amount of land taxes increases, the budget to make the loan payments decreases, which should reduce the property value (since the same logic should apply to all the potential purchasers).
It gets more complex once you add land speculation but I think the core logic remains the same - instead of being willing to pay $2000/mo for the property the developer might be willing to pay $2200 anticipating a speculative land return (without LVT, so higher property values), but that shouldn't change in general the market rental rate or if construction is viable, so I think the main difference with or without LVT would still be the land values.
Well, LVT is always meant to be a replacement for conventional property taxes so depending on how it is implemented it can be roughly the same as the previous property tax or it could be significantly lower because only the land is taxed and not the whole property.
You would be describing parts of Detroit wherein land is abandoned and the city limits shrink as the cost of providing services increases.
Institute an LVT and those competing landlords may become cashflow negative, forcing some of those units off the rental market entirely (typically to sales to owner-occupants, sometimes to landlords with better financing or lower overheads, sometimes to short-term rentals or other higher cashflow uses).
An extra $100/mo of LVT may not result in an extra $100/mo of market-clearing rent, but it's far from obvious that it results in $0/mo change.
Thought experiment: if the LVT for that unit were $3500/mo, what do you think the offered rent would be? If that's the case for $3500/mo, what part of that doesn't hold for $3000/mo, $1000/mo, $350/mo, or $35/mo?