I've used newspaper logs, and, while they can stretch your wood supply, they won't substitute for it altogether. Paper burns with a lot of ash because it doesn't burn as hot as wood when it's compacted, so you need wood to burn with it. If you usually use say, three medium pieces of wood, you can use one or two of paper instead. It burns a little better if you soak it in water first then give it a couple of weeks to dry out.
I saw the paper brick maker and was intrigued - wish I still had a wood burning stove. It might burn better as a brick because it would be compressed much more than logs.