Would having all housing operate at non-profit rates mean that any increases in a Land Value Tax burden mean that it increases rents?

Conventional wisdom regarding the Land Value Tax states that it cannot be passed on to tenants of a residence, due to the fixed supply of land; meaning that only the one who actually owns the land would be burdened from the tax. But, to my knowledge: This is assuming a for-profit free market system.

Wouldn't this logic then fail to apply to a housing market in which all supply were operated at non-profit rates? Or would this still apply, but the driving factor would shift from "profit maximizing" to "tax burden minimizing" (or maybe something else)?