What stops firms increasing prices of goods in the case of an increase of a direct tax such as corporation tax?
The difference between direct and indirect taxation is not that the first cannot be passed onto a consumer. The difference is that in case of direct taxes the direct tax is imposed upon a person and not a transaction (there is a convergence between the source of taxation and the entity subject to taxation).
Direct taxes are paid by a entity subject to taxation, but can be passed:
onto a buyer (consumer) of the taxpayer (shifting the tax ‘forward’), bo also
onto a seller to the taxpayer (shifting the tax ‘backward’).
Of course, the taxpayer can also cover the increase of taxes by lowering its profit.
For example, if a convenience store is subject to increase of taxation, it shouldn’t pass it onto consumers, when the consumers have access to a cheaper substitute (eg. grocery store not in the scope of increase of taxation). On the other hand, if the market between store’s suppliers is competitive, the convenience store can pass the increase of taxation onto suppliers.
In case of an indirect tax, there is always an increase in price. Why would the seller want to increase it further beyond the equilibrium point.
Also, direct taxes theoretically are not meant to be passed on to the consumer since those are performed by indirect taxes. Direct taxes like income tax are specifically meant to tax people's purses.