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/