If your burner was designed to run on meths (ie alcohol) it is not likely to perform properly (unmodified) on paraffin. Alcohol as a fuel needs a lot less air than paraffin, which is why paraffin will produce smoke – incomplete combustion and soot and carbon monoxide in the smoke.
There is no likelihood of a problem from a reaction with residue meths. Parffin is so called because it is fairly inert, chemically (parvum affinitas if my O level latin (failed) is any good).
There are (or were) stoves that would run well on paraffin, for greenhouses etc, or you could perhaps find a pressurised stove (Primus, Optimum, etc) or blowlamp which run very well on parafin with no smoke, but a lot more heat than an 'ordinary sized' Stirling engine can cope with. I think.
Cheers, Tim