No, because the source of the Sun's energy is not from ordinary combustion but from thermonuclear fusion deep inside its core, at temperatures of millions of degrees. The Sun, like many other stars, does produce a 'wind' of particles which travel through the solar system at about 450 kilometers per second. This wind consists of electrons and protons in a plasma, laced with a weak magnetic field, at a density of about 5 particles per cubic centimeter.
Some, very old stars in their red giant or supergiant phase such as Betelgeuse, are only a few thousand degrees hot in their outermost regions. Under these conditions, carbon and silicon atoms can combine to form dust grains and 'soot' which get blown out into space by radiation pressure. In a manner of speaking, these stars are producing 'smoke', but by a process of condensation not combustion.