ArcGIS will pop-up a warning message when the campus license for ESRI products will expire within 15 days. This serves as a notice to the license manager at the GIS Facility to renew the license file; it does not require action by anyone else on campus.
If the message is annoying to you and you don't want to see it anymore, on your machine go to C:\Program Files\ArcGIS\Utilities and double-click the AdvancedArcMapSettings.exe file.  Open the Miscellaneous tab and uncheck the box on the right side, 1/2 way down regarding License timeout warning.
Then click Apply.