That question was raised last week in a scientific paper by Adam Stevens, Duncan Forgan, and Jack O'Malley-James. They're really interested in the opposite question: if a distant civilization destroyed itself, would we see any evidence of it? This question arises because one of the answers to the Fermi paradox (why don't we see any other civilizations out there?) is that advanced civilizations tend to destroy themselves, either accidentally (oops, I dropped my hammer on that big red button) or on purpose (I've genetically engineered a virus that kills only annoying people....)
Stevens, Forgan, and O'Malley-James look at a variety of ways that we might obliterate ourselves. My takeaway from their paper is that it would be surprisingly hard for aliens on a distant planet to detect any evidence of our self-destruction. For instance, if we simultaneously detonated every nuclear weapon on the planet (I am not recommending this, by the way), the resulting gamma-ray signal would barely be detectable at the edge of the solar system. The main way that we might be able to see evidence of another civilizations's demise is spectroscopic: through a change in the chemistry of the atmosphere from a nuclear war, or the release of gases from the death of every living thing on the planet. (Are you starting to get the impression that this paper is a bit of downer?) On a happier note, it's exactly this sort of spectroscopy of the atmospheres of distant planets that could give us the first "smoking gun" evidence for the existence of life on other worlds.