Is it cheaper to heat a building burning wood?

The reality is if you can find the wood free it is cheaper for heating purposes. But you will probably have to cut it, stack it and cart it in and dry it.
The reality is if you have a gas fire next to a wood burner you will light the gas fire as it is quicker and less work and you feel the heat straight away. The wood burner becomes a white elephant.

Then there is the work of cleaning up a wood burner afterwards. The initial cost of installation is often higher for wood burners.

The romantic angle of wood burners is gone now too, as there are gas flame effect fires that look and feel just like a wood burner. And some are efficient too at around 90% or more.  Now if its romance you want get us to install spa bath big enough for two.

Evolve-1250-Gas-Fire
Evolve 1250 Gas fire
simulates a real open gas fire. There is an option for an extra glass grill protection.

The main reason for installing a wood burner is that natural gas is not available and you have a big area to heat. There is no replacement for an open wood fire if thats what you want  but remember that all that heat going up the chimney is dragging cold air into the house also.

When the weather is really cold heat pumps lose their efficiency and the cost goes up. Also when your wood is wet less heat is given out as most of the heat is spent drying the wood.