Burning wood emits smoke and black. Provided more oxygen or whatever required, can wood be practically burnt fully like petroleum gasses that emits a blue flame and little smoke and little black.
Burning wood is three processes:
- Gasification - under heat and little oxygen, the wood is turned into combustible gases (mostly Carbon Monoxide, Hydrogen and gaseous tar) and charcoal
- combustion of charcoal
- combustion of gasses
The optimum conditions for these are not exactly the same (I'll dig out my thesis to look up the particulars). A good stove will have one area where char burns and the gasification happens. Then, additional air is added to the (combustible) exhaust to burn it more or less completely.
An additional complication is that thegaseous tar tends to (partly) polymerize in the flame and form soot that usually does not burn - the yellow in the flame you see is glowing soot.
So, in summary, burning wood completely is sometimes possible but hard, but good approximations exist.
Wood is mostly cellulose, which burns to yield carbon dioxide and water, so this indeed can "burn fully". The rest - particulate matter in smoke and wood ash is made of minerals that the tree either drew up from ground water and could not dispose of or stockpiled if they are biologically useful. So much of the rest is pretty much non-combustible anyway. See the Wikipedia page on Wood Ash where it is stated that the remnants include things like calcium carbonate, calcium oxide and iron oxide.
Many of these would vapourise if the wood is burnt in a reverberating furnace with forced draught (plenty of oxygen), but this is likely not quite what you mean by "burning".
So it seems that if the wood ash contains no charcoal (i.e. carbon, which can burn at higher temperatures to yield more energy), and is the fine white powder that you shovel out of a hearth, then the wood has pretty much burnt as much as it is going to.