Yes, you can treat his possessions (including the dog, which is according to law just a possession of personal property) as "abandoned" if you've acted reasonably to give this roommate/tenant notice of a deadline by which you're going to get rid of their property.
Think of what would happen if the roommate came back and found you had gotten rid of their dog and other stuff --how would you defend it? You'd show the court a copy of a letter sent by a provable means of delivery - FedEx, registered rPost email, certified US mail) to all their known addresses, families' address, etc., warning them of your deadline, and their lack of response. That way, they couldn't claim you didn't try to reach them and give them a chance to respond before you disposed of their property.
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I agree with my colleague. You can contact animal control and tell them your roommate has abandoned his dog. They may be able to help.
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