Burning the manure takes out the rich source of organic matter that could help your garden retain water and other nutrients. It is better to compost it and then use it for fertilizer. Making those manure logs seems like a nice idea, but it is really labor intensive. Also, I just don't think that my DH would go along with turd bricks by the fireplace. I think that would take a lot of the romance out of the whole fire thing.
I think both methods, burning then adding it to the top of a compost pile would be a win/win situation.
If you were using the heat from burning to heat your house, then it would probably be a good thing. If you were just doing it to get rid of it, I think it would be a lot of smoke, a lot of trouble, and a lot of mess for nothing.
I used to pick out my fields every day... it was about an 30 min each paddock and would compost it. I had 3 paddocks on about 9 acres and was lucky they would always go in the same place so never had to walk forever around the whole thing! Lol.. then would hit the stalls... I wonder over the years how many miles I logged in!