First of all, I think "many" people believe the rich are not paying their "fair share," because it is a politically engineered sentiment. The more the government can get support from the majority to tax anyone more heavily, they will do it. Government's natural inclination is to grow in size and power, and taxation allows this to happen. I do not think that Americans, who all aspire to be wealthy more than individual citizens of other countries, actually sit back in their easy chairs seething with jealousy and class warfare. Perhaps some do, but I don't think that is the majority. I think most Americans believe people ought to be as economically "free" as possible and that heavy taxation hinders this freedom.
Secondly, let us not forget that the discussion about taxing the "rich" is a political distraction, designed to get our attention away from the real issues. The math tells us as much. When it comes to getting the government the revenue it needs (re: balancing the budget), income taxation would have to rise on ALL the top earners (those making over $180,000 per year) to a rate of 88% (not marginal but actual). This would leave all households within this bracket worse off than the households at the poverty line, which is untenable. So this means you would have to double the income tax rates (not marginal but actual) on everyone making $100,000 a year or more. This means a household with two wage earners, each making $50,000 a year, would pay an additional $21,000 a year in taxes. This is also untenable.
The point is that government spending has to be talked about, and this will lead to discussions about the proper role and function of government in society, which are politically risky topics. It is easier to make a straw man argument about greedy rich people.