The corporate income tax encourages firms to use debt finance, rather than equity. Debt finance makes companies riskier. But because payments on debt are tax deductible, and dividends are not, companies have a strong incentive to use debt rather than equity finance. The deductibility of debt payments also lowers the required rate of return for new projects, possibly encouraging companies to invest in marginal ideas that aren't really worth it. Without the corporate income tax giving them a 35% reduction on their interest payments, they might think twice.
The corporate income tax encourages firms to waste resources on tax avoidance. In general, taxes are most efficient when they fall on those who have the most difficulty avoiding them. Big corporations can and do spend an enormous amount of money and human effort transforming their income into more tax-preferred forms--deferring it, moving it, swapping it with entities that have different tax rules, and so forth. We spend an enormous amount of energy trying to make rules to stop them. It would be a lot easier to get rid of the thing entirely and focus on getting the money from people, who can't afford quite such large squads of tax attorneys. This would also correct an obvious flaw in the corporate tax code: it's easier for big companies to afford pricey tax lawyers--and pricey lobbyists to get them special tax breaks. Moreover, as I hinted above, the rules governing corporations are complicated for a reason--what constitutes an expense is as much art as science, and varies from industry to industry.
Without the corporate income tax, a lot of the incentive for lobbying would go away. Not all of it, by any means--I am not trying to paint some halcyon future here. But an enormous amount of effort goes into lobbying for tax laws, and politicians often reward favored constituent businesses with little sweetheart fillips to the tax code. Conversely, apparently neutral changes to the tax code often turn out to be excellent ways to hamstring your competition, particularly small businesses who cannot afford a huge tax department.
With equity, financiers share in the risk, and the moral element of the shift from equity to debt financing is widely under-appreciated. Similarly, the mere presence of government, in taxation or regulation, provides opportunities for regulatory capture and outright corruption. Removing the opportunity for corruption is a far more effective means of draining the swamp than any enforcement could hope to be.