I'm a frequent visitor to Turkey and think it is one of the very best destinations in the world. That said, I would not blow off this situation. In the recent past, Kurdish extremists have targeted tourist sights in Turkey and foreign tourists have died in bombings. I would be concerned about an escalation of tensions that could result in similar incidents. Frustration and being locked out of the political process are a recipe for increased likelihood of violence.
And I'm not sure that I would trust the judgment of a hotel staff. True, they are on the scene and can give you a sense of what they are seeing and hearing on the street. But unless they are plugged into the extremist community, they might be as surprised as anyone when something happens. Also, they have a strong financial incentive to downplay the possibility of violence, lest the hotel be empty of paying customers.
Frankly, as much as I love Istanbul, I would not travel there myself at the moment. The odds, of course, are overwhelmingly against being caught up in sectarian violence, but times of heightened tensions are when those odds do undeniably go up.