The unfair competition rule that you have heard about is probably the tax on unrelated business taxable income imposed by Congress many years ago to discourage competition with for-profit businesses.  Unrelated business income tax (“UBIT”) is imposed on the net income of a tax-exempt entity from any trade or business which is regularly carried on and unrelated to the purpose for which the organization has been recognized as exempt. An art museum is not likely to be taxed on the income from selling books on art, but is likely to be taxed on the income from selling racy novels.

In your case, if the organization has accumulated a lot of costumes, presumably at least one of its purposes is the promotion of theater in the school.  Whether that purpose is sufficiently broad so as to encompass promotion of theater at another school is something I can’t tell from here.  It doesn’t really sound like it, but it depends significantly on your governing documents and your Form 1023 application for recognition of exemption.  It could also depend on whether you can intentionally broaden your purpose to help other schools.

Even if it is an unrelated activity, however, it is still not subject to tax unless it is “regularly carried on.”  Renting costumes on a one-time basis would not be regularly carried on, and any net income from this one inquiry would not be subject to tax.  If you did it frequently, or as part of a general practice of renting or selling stuff to others, it might well be considered part of a trade or business regularly carried on.  (See Ready Reference Page:  “Nonprofits Often Worry About UBIT”)

If you are interested in doing something like this more than once, you should consult with an attorney or accountant familiar with the UBIT rules to see if you can structure the activity so that it avoids UBIT.