Does a new landlord have to carry over the tenant's last month's rent deposit after a sale in Ontario?

Yes. A last month's rent deposit held under the Residential Tenancies Act, 2006 belongs to the tenancy, not to the individual who happened to be landlord when it was collected, so it needs to transfer to you as the new owner rather than disappearing with the seller. In practice, this is handled as part of closing: the seller credits the deposit amount to you through the statement of adjustments, so you effectively receive the funds at the same time you take on the ongoing legal obligation to hold and eventually apply it.

Once you own the property, you become responsible for that deposit going forward, including applying it toward the tenant's final month of rent when the tenancy eventually ends. Before closing, confirm with your lawyer exactly how much deposit is being held, that it matches what the tenant understands is on file, and that the credit is properly reflected in your closing figures, since a mismatch here is a common source of dispute after a buyer takes over as landlord.