So if corporate tax was increased, I have to ask what stops the prices from increasing for the consumer?

Nothing stops this.

In fact, most of the burden of corporate taxes probably rests with the consumer, not the corporation.

The economic burden of a tax is always split between buyers and sellers. If sellers are taxed, they raise prices. If buyers are taxed, they punish the seller by buying less of the seller’s product.

The split depends on the proportional elasticity of supply and demand. The more price-sensitive you are, the flatter and more elastic your supply or demand curve is. The less price-sensitive, the more vertical and inelastic.

If you have price-sensitive demand, then you are willing to abstain from buying the good if it is taxed. So the seller has to eat the tax out of their own profits. If the buyer is taxed, the seller might have to cut the price in order to refund the buyer. And if you have price-sensitive supply, then you are willing to stop producing the good at all unless the buyer agrees to pay for the tax.

For example, suppose liquor stores were taxed. A liquor store has elastic supply because they don’t care whether they sell liquor or soda. But people who drink alcohol cannot drink soda instead. So if the liquor store is taxed, they will basically say, “We will raise our prices enough to pay for the tax, and if you buyers think the price is too high, we’ll just become a 7-11 and sell chips and soda instead.” The alcohol drinker needs alcohol more badly than the liquor store does, so the burden of the tax will be borne mostly by the drinker.

In general, corporations have relatively elastic supply because they will sell literally anything that makes them money, and they don’t care what specifically they sell. Meanwhile, customers have relatively inelastic demand because we want what we want. Customers care more about the product than sellers do. E.g., I need a car more than Ford needs to sell me a car (since Ford could make airplanes or agricultural tractors or bicycles). So we should expect that relatively speaking, most of a corporate tax will be paid by consumers.