In short, it would be a nice idea. But I'm firstly waiting for a certain someone to come along with his trademark "Requested, discussed, and rejected" line, and proceed to make people feel stupid for even suggesting the idea in the first place. The second would that as I mentioned right at the start, it could be complicated. But a whole lot of fun if it devolves into some kind of rebellion or civil war, I suppose.. 
Actually, I think that's the first time I've heard that suggestion
However, I believe that the intent behind protests is that everyone who
doesn't protest is already implied to be supporting the person being protested against.
If we were to implement something like this, it would probably require rebalancing the number of protest "votes" needed to boot someone out of office.
