What are the risks of buying a rental property where the tenant pays cash and has no written lease?

The Residential Tenancies Act, 2006 generally applies to a residential tenancy whether or not there is a written lease and regardless of how rent happens to be paid, so the absence of a written agreement does not mean the tenant has fewer protections, or that you as a buyer would find it easier to end the tenancy. The same notice requirements and grounds for ending a tenancy still apply.

The real practical risk is the lack of documentation itself: without a written lease, you have no reliable record of the actual rent amount, when the tenancy started, what was agreed to, or whether a deposit exists, all of which makes both your due diligence and any future dispute harder to navigate. Before closing, ask the seller and, where possible, the tenant directly to confirm these details in writing, since getting even an informal written confirmation now is far easier than trying to reconstruct the history of an undocumented tenancy after you have already taken over as landlord and a disagreement arises.