Corporate taxes come out of profits so prices don't rise (because of all of that wonderful free market competition) unless the corporation has a monopoly (which the government should prevent).
Nope, they don't come out of profits. There's a return rate that corporations have to achieve to attract investment, to have good bond ratings, etc. What this rate is varies by market segment (roughly correlated with segment risk), but there's also a sort of overall norm; companies either provide that rate of return or they fail, get bought out, split up, etc.
When you increase taxes you don't change any of that, you just force companies to adjust. In the short term their after-tax profit margin will take a hit, but within a few years it will be back up because they (all of them, collectively) will have adjusted. Wages will be pushed down, suppliers will be paid less, prices will be pushed up... but the profits will be back in line with expectations. Note that this also applies if you cut taxes: in the short term profits will rise, but competition will squeeze them back down to the expected rates.
I'm talking about long-term, steady-state equilibria here, of course. In practice it's more dynamic than that as innovations or particularly good or bad management boost one company or damage another, and there are market-wide periods of growth and decline, but overall, and over time, that's how it works: Companies will generate the market-demanded rates of return, or they'll be defunded in favor of others that do.
A great primer on this, BTW, is Thomas Sowell's "Basic Economics". Be prepared for the fact that Sowell is a conservative, but if you can look past that he provides beautifully simple and lucid explanations of market dynamics.