Do I inherit responsibility for property taxes the previous owner owed when I buy through power of sale?

Generally, no - outstanding property tax arrears are typically dealt with through the same statement of adjustments process used in any Ontario closing, rather than becoming a personal debt you take on as the new owner. At closing, the lawyers involved would normally confirm the amount owing and adjust or pay it out of the sale proceeds, so the taxes are addressed as part of the transaction itself rather than quietly following you afterward.

That said, power of sale transactions are still worth extra caution here specifically because the usual cooperative back-and-forth between buyer's and seller's lawyers may be more limited when the seller is a lender rather than an engaged homeowner. Confirming the tax account is genuinely up to date, or properly adjusted for, at closing is a step your lawyer should not skip simply because the transaction is otherwise moving quickly.

Ask your lawyer to specifically confirm the property tax account status as part of closing, rather than assuming it has automatically been handled the way it typically would be in an ordinary sale.