No, they'll lower the prices, just like how they raise prices when taxes go up. You'd also have to raise the sales tax a bit. The idea is to switch the tax system over to a non-avoidable method.
But the one problem is that sales tax is not as fine a tool as income tax. Still, it's worth it.
While this intuitively may make sense, I don't think real world experience bears this out. This past summer, FAA authority to collect taxes on airline tickets expired. As a result, airlines no longer needed to collect taxes on the sale of tickets (which can be quite substantial) for period of weeks. Rather than lower prices, every single carrier, without exception, pocketed the difference, rather than lowering prices (the particular taxes in question are built into the cost of airlines tickets, they are not the fees tacked on at the end).
I disagree with eliminating corporate income taxes, as it would only serve as an encouragement for corporations to retain income, rather than investing it in their business (as doing so isn't taxed - only their income after expenses is taxed). I do agree, though, with substantially lowering corporate tax rates, while eliminating deductions. We have high corporate taxes because of the ridiculous loopholes in the system, which generally favor massive multinationals over the small businessman. Reduce the tax substantially, tax all companies equally (but keep it revenue neutral) and that's a plan I can get behind. The problem is all the special interests defending their individual tax loopholes. In any event, though, I think majorities in both parties realize we need to reform the system.