What recourse if a company offers a promotion and refuses the pay increase after performing the job duties?

If you had an employment contract or agreement detailing the additional pay you'd receive for accepting the promotion, you could sue to enforce it. In the absence of a contract, however, the law does not enforce promises in the vast majority of cases; the employer would be free to renege on its promise of more pay. You could potentially enforce the promise only if you could show that you had to do something significant to your detriment to take the promotion, and the company knew you'd have to do that, and knowing that, made you the offer anyway.

Simply working harder or having more responsibilities, however, would not be a sufficient detriment to enforce the promise, since your employer, in the absence of an employment contract delineating your duties, could change your duties, give you more responsibilities, or simply make you work harder any time it wanted you to. (Without a contract, you are an "employee at will," and your employer may do anything it wants vis-a-vis your employment, up to and including terminating you.) Rather, the sort of detriment which would be required to potentially enforce this promise would be something like accepting a transfer to a different (and distant or less-desirable location); giving up some benefits to take the promotion (e.g. giving up the ability to earn overtime); accepting a transfer from day to night shift; giving up some other job offer; etc. Without your having to do something of that scale to your detriment to take the promotion, you would not be able to hold your employer to its promise.