I haven't done this for running-specific purposes, but I have had a toenail surgically removed when it blackened.
Previously I had found that burning a hole in the toenail over the blackened area relieved the pain and allowed the accumulated blood to escape (I had a doctor do this, but it is easy enough to do with a needle and gas range). Once, after doing this on my big toe, it became infected and was oozing puss out the hole. The doctor said I should have the toenail removed (risk of deeper infection, gangrene, etc.).
The procedure was among the most painful things that I have ever experienced. The injection of topical anesthetic into a digit, or any confined area, results in massive pressure. My toe looked like something out of a Tom 'n Jerry cartoon. It also took a handful of days to heal and in the interim my running was limited.
All in all, I cannot see this being worthwhile -- toenails, while a vestigial product of evolution, really don't get in the way too often.