Case:
I was hired to perform specific tasks as an exempt salaried employee--Help Desk/Desktop Support. Shortly after being hired additional duties are added that would have been the job of another employee for which the company decided not to hire. I must add the other position was a higher technical job that paid almost twice my salary--Network Administrator.

After confronting management about due compensation for the additional duies and basically working two jobs I was told no increase. There is a line that states additional duties as assigned in my offer letter. I do not believe additional duties can include that of a second job just because they elected not to fill the other position

My question is what recourse does the company have if I elect to stop performing these additional duties based upon my original offer sheet and industry standard for the job I ws hired for?

Of course, they can. Unless you have a bona fide contract that limits what tasks you can be asked to do, you have no recourse except, it appears, to look for another job. They have already said that they are not prepared to adjust your salary at this point. If you refuse the additional duties, you can be fired. You don't get to determine the job description and what you will and won't do. Unless they are asking you to do something illegal, what they are doing is perfectly within the law.