It’s a lot more complicated than that with a mix of short and long term effects. It should make home ownership more desirable, but taxes don’t destroy wealth so lowering income taxes for example could largely offset any increases in rent.
Further the value of property would decrease, but the cost of empty properties would increase. That would put downward pressure on rental costs.
That said I would support a hard progressive tax on non-primary residence homes. It should really hurt for them to own multiple homes.