Is incineration of a solid fuel complete or incomplete?

This depends entirely on the conditions.

If there is enough oxygen to make the mixture stoichiometric or oxygen-rich, and it burns for long enough at a high-enough temperature. However, the precise meaning of each of those three "enough"s will change with the material, so it is impossible to give a definitive answer for a non-specific case.

As a general rule, the darker the ash produced by the combustion, the less complete the combustion process, as a darker color is typically caused by remaining charcoal (unburned hydrocarbons). If you start with biomass, it will consist of a lot of hydrogen (which forms water and evaporates), carbon (which, if completely combusted, forms gaseous carbon dioxide), nitrogen (which forms gaseous N$_2$ or NO$_x$) as well as a sundry mix of other elements, which will cool down into mineral oxides in the ash. What that will look like will depend on what those added elements were in the initial biomass.