How can I make an animal like a dog disappear and make people think it really disappeared?

The easiest way to make people think that an animal has disappeared is to make it actually disappear. Many household animals (particularly cats) will run away if you just open the front door a little. Other animals can be hidden in basements, tied up in bags, sold to disreputable pet stores, or surreptitiously given to a small child on the street.

The problem with these methods is that they a) can get you in trouble with animal protection agencies, and b) may make life difficult if you want the animal to reappear on cue at a later date. But if you just want an animal to disappear, they work fine.