Even though the following problem is usually not listed as a typical OR-problem, its solution might (depending on how the solution looks like) have a HUGE effect on OR algorithms. I am speaking of nothing less but the
P vs. NP-problem, which is one of the most important open problems in all of mathematics (and computer science). It is also listed as one of the Millenium Problems and its solution would bring the person to solve it $1,000,000 from the Clay Mathematics Institute.
The problem ask whether the two formal classes P and NP are the same or not, i.e. whether P=NP or P$\not =$NP. While the inclusion P$\subseteq$NP is rather trivial, the other inclusion is not. In fact, it is widely believed nowadays that P$\not=$NP is true (c.f. the latest P =? NP Poll).
If, however, it would turn out that P=NP, then this would have a tremendous effect on OR because this would imply that all those NP-complete problems which people from OR like to deal with (TSP, Graph Coloring, Integer Programming in general...) would allow polynomial-time solution algorithms.
If the solution were P$\not =$NP then this would not have such a big effect on OR as P=NP. Yet, for people working in OR it would nevertheless be interesting to know the final solution to the P vs. NP question.