Does a buyer inherit an already-filed LTB arrears application against a tenant when they buy the property?

Generally yes, in the sense that an already-filed application and its hearing process continue regardless of the change in ownership, and the new owner typically steps into the previous landlord's role for the purposes of that proceeding. This means the file itself does not disappear or need to be restarted just because the property was sold partway through.

The right to actually collect any arrears that accrued before closing, however, is a separate question from the pending application continuing; that right generally stays with the seller as the landlord during the period the arrears accrued, unless the purchase documents specifically assign it to the buyer. Because a pending application and the underlying right to the money it seeks are not automatically the same thing, this needs to be addressed explicitly in the closing paperwork, spelling out who is entitled to any amount ultimately recovered and who is responsible for participating in the ongoing proceeding. Confirm this clearly with your lawyer before closing rather than assuming it works itself out.