A dog can be thrown away in the trash such as in a public dumpster or in your own trash can at home if the dog will fit in the trash can.
You can put the dogs body in a black trash bag and then place the dog into your trash can and when the trash service comes to dump your trash can the dog will just go into the trash truck and be compacted.
Then the dogs body will be taken to the dump or transfer station and be buried along with the other trash.
If you can though it's best to dig a hole in the ground at least 3 feet deep and bury the dog instead of throwing it away.
But sometimes you may not have any other options.