Does protecting US sugar producers
drive up the price?

Yes, but compared to what? Nearly every country in the world protects, subsidizes and micromanages the details and economics of sugar commerce within their borders, so the US situation is just one among many. There are only a handful of countries in the world that allow true, unregulated supply-and-demand sugar capitalism. And in spite of US "protectionism," Americans pay less for sugar than most industrialized countries, and in a recent study, only Australia and Canada had lower average prices.