Does a property owner owe a duty of care to a trespasser who is injured in Ontario?

Yes, but a reduced one. The Occupiers' Liability Act recognizes that the duty owed to someone on a property without permission is narrower than the duty owed to an invited visitor - generally, the occupier must not create a danger with the deliberate intent of harming the trespasser, and must not act with reckless disregard for the trespasser's presence and safety, rather than being required to proactively make the entire property safe for someone who was not expected to be there.

This lower standard reflects the practical reality that an occupier cannot reasonably guard against every possible way an uninvited person might enter a property and be hurt. That said, the reduced duty is not the same as no duty at all, and specific circumstances can still affect the analysis - for example, an occupier who knows a particular route is regularly used by trespassers, or who maintains a particularly dangerous hidden hazard in an area people are known to access, can find that this knowledge affects whether their conduct amounts to reckless disregard in the circumstances, even under this lower standard.