There have been theories that our solar system has a hidden dead star (Nemesis) far beyond the Oort cloud. Such hypothesis generally assume the star is now in the form of a red or brown dwarf due to the limiting constraint that we cannot see it. And realistically, even if it were still burning it would probably be too far away to look like a "second sun" without it being too massive for normal sun's planets to be stable. More likely it would be seen as a very bright star.
But even then it would have pretty bad implications for the stability of the system's planets. That's in fact one of the things that led to the Nemesis hypothesis; apparently periodic extinction events that could be explained by periodic meteor bombardments caused by a distant massive object mucking with the orbits of comets and sending them to the inner planets.

There have been theories that a hidden, dead star (named Nemesis) orbits our solar system far beyond the Oort cloud. Such hypotheses generally assume that Nemesis is now in the form of either a red or a brown dwarf due to the limiting constraint that we cannot see it. And, realistically, even if it were still burning, it would probably be too far away to look like a 'second sun' without being so massive that it would destabilize our main sun's planets' orbits. In all likelihood, Nemesis would be seen as a very bright star.
But such a star would, even then, have pretty bad implications for the stability of our solar system's planets' orbits as previously stated. That is, in fact, one of the things that led to the 'Nemesis' hypothesis; apparently, a periodicity has been been observed in the frequency extinction events, and this could be explained by periodic meteor bombardments caused by a distant massive object mucking with the orbits of comets and sending them spiraling in toward the inner planets.