I don't think real analysis comes into play much if at all on the modeling side. When it comes to designing algorithms (or understanding how they work), it may be helpful for nonlinear optimization (as Siamak mentioned), and it almost surely would be helpful in understanding optimal control or calculus of variations problems. I tend to associate those more with engineering than with OR, but your mileage may vary.

That said, and bearing in mind that I took real analysis both as an undergrad and in my PhD program, let me add this: regardless of whether you'll ever apply real analysis, the process of learning it (and of meeting whatever challenges it may pose for you) may pay dividends both in grad school and in your career in OR. Mental faculties are not all that different from physical ones; you gain more from working at what you are not good at than from working on just what you are good at.

You need a solid background of real analysis specially when you want to study or use nonlinear optimization. Any effort you put into understanding real analysis today will be paid off later :)

Real analysis will also be necessary if you are planning to do research in Game Theory and also Probability and Queuing Theory. I totally agree with Paul, if you can develop a taste for real analysis, it will pay well.