If you are talking about smoke from biomass combustion, the main residual inorganics (some are not metals) are Si, K, Cl, Ca and non-graphitic but relatively chemically inactive soot. If the biomass combustion is especially poorly done (wet material, poorly ventilated, heavy smoke emissions) then the smoke/dust will also contain creosote, tars, and other materials. Si from some forms of biomass (straw and some grasses) can lead to serious lung diseases. The creosote and tars are also special health concerns. The K and Ca are less of a concern. Cl is always a concern. 

Household dust that does not derive from biomass contains inorganics common to dirt (Si, Al, Fe, Ca, Na, etc.). It also contains a surprisingly high amount of organic material in the form of pet and human hair, dead skin cells, and, in the kitchen, smoke from cooking (soot, oils, aromatic and non-aromatic tars, etc.). 

For biomass burning in household furnace, the trace metals contained on the the surface of dust is various, which is depended on the types of biomass ( straw/woody/agricultural biomass, etc), growing soil, growth cycle, fertilization times, etc.. Except from Si, K, Cl, S, Al, Mg elements, suggest you could pay attention to the Cu, Zn, Ti, Fe elements. 

As Larry pointed out metals are likely to be one of the least significant health threats from biomass burning in a domestic setting. The incomplete combustion of organic materials can produce poly aromatic hydrocarbons (PAHs) some of which are carcinogenic (testicular cancer was an occupational disease associated with chimney sweeps in post industrial revolution Europe). The particulate - particularly fine particulate matter is also a concern as it gets into the deepest part of the lungs and if this also carries organic pollutants, these can be readily passed into the blood stream at this point. The dust from burning will contain metals (those adsorbed by the plant material), but unless the ground in which it is grown is contaminated (road sides, high use of phosphate fertilisers with cadmium residues, poultry manure from animals fed with copper suppliements etc), this may not be much more than background from urban inputs.