Does a buyer assume the seller's obligations for a tenant's deposit and arrears when a rental property sells?

For a last month's rent deposit, generally yes; Ontario's rules do not typically permit a general security deposit beyond a last month's rent deposit and, in limited circumstances, a key deposit, and whatever last month's rent deposit is actually being held transfers to the new owner along with the ongoing obligation to hold it and eventually apply it, usually with a corresponding credit to the buyer at closing.

Arrears are a different matter. The right to collect rent that was already owed before closing generally stays with the seller as the landlord during that earlier period, unless the purchase documents specifically assign that right to the buyer, while the new owner becomes responsible for collecting and dealing with anything that accrues going forward. Because this split between deposit obligations and arrears rights is easy to overlook and can lead to disputes if left unclear, confirm exactly what deposit is being transferred and how any existing arrears will be handled directly in your closing documents, rather than assuming it works itself out automatically.