I'm in central Istanbul at the moment, and other than a few areas near Taksim Square, the protests are easily avoided. The Old City is completely unaffected.

I expect it's a bit like the riots in London in 2011: unless you're in an affected neighbourhood, you experience it from watching the news. The police here are quite liberal with their use of tear gas and water cannon, so as long as you avoid the hotspots, it's no less safe than normal.