Here are a few institutions that host student design competitions.
CMYK
CMYK is a great publication aimed specifically at students. It hosts a variety of competitions.
http://www.cmykmag.com/
Print Magazine
Print Magazine hosts a student cover design competition each year.
http://www.printmag.com/competitions/tabid/80/Default.aspx
Communication Arts magazine
Communication Arts magazine has specific categories for students within Photography and Illustration but the design competition requires client based work.
http://www.commarts.com/competitions
How Magazine
How Magazine allows for student entries in all of its major competition categories.
http://www.howdesign.com/competitions/
