A property tax:
In the short run, a property tax will have no effect on rent, as the supply of homes is fixed (i.e., supply is totally inelastic in the short run), and demand is relatively inelastic but not completely so, so the incidence will fall on property owners. In the long run, a property tax will increase rental costs because supply of new units (i.e., construction) is highly elastic and long-run demand for housing looks just like short-run demand— relatively inelastic. As a result, the incidence of the tax will fall on occupants (both renters and owner-occupiers).
Intuitively, one could think of it this way: the cost of ownership will increase due to the tax, so for someone to see it as worthwhile to buy a newly-constructed home to rent out, the rental income would have to be relatively higher in the presence of a property tax than if one were not in place.
A tax on rental income:
The tax will increase rent; it will decrease the supply (and demand) of rental units (and increase the supply and demand of owner-occupied units). This will happen because the tax will reduce the return on apartments as revenue assets, causing some number of landlords to sell them to owner-occupiers, taking them out of the rental market. More importantly, though, there will be some renters who do not have the option of switching to owner-occupancy, and they'll bear the incidence of a tax on rental income.
I am assuming that there are three populations:
- Landlords, who can choose to purchase homes for rent, but live elsewhere, so their choice to buy a unit to rent out can be analyzed separately from their own housing consumption
- Renters, who must rent, due to some frictions (inability to afford a down payment, poor credit, unstable or difficult-to-document income history, etc.)
- Potential owner-occupiers, who can choose between owning their own home and renting from a landlord, and who have some set of preferences between ownership and rental
Thus potential owner-occupiers and landlords compete for units for sale, while renters and potential owner-occupiers compete for rental units.
In this case, the choice of landlords to exit due to taxes will not pass through perfectly to the rental market due to the fact that while potential owner-occupiers who are currently renting can escape the tax by switching to ownership, there are some renters who are renting whose demand for renting is pretty inelastic— these people will bear the incidence of the tax.
Interestingly, in the US there is such a tax on rental income, though it's implicit. We have a bunch of incentives for home ownership, notably the exemption from capital gains for owner-occupied homes, but most importantly the fact that income on rental properties is taxed like any other business income, while owner-occupiers effectively rent their homes to themselves without paying tax on the implied rental income. (As an aside, people often point to the income tax deduction for mortgage interest payments, but this actually just creates tax equivalence between homeowners and landlords, who can deduct interest as a business expense.)