Here is a list of writing contests for high schoolers that I have accumulated for a while now (they include poetry, nonfiction, short story, essay, and other categories):
Poetry:
http://arts.princeton.edu/about/opportunities/high-school-contests/poetry-contest/
http://www.justpoetry.org/home
https://www.hollins.edu/academics/majors-minors/english-creative-writing-major/nancy-thorp-poetry-contest/
http://www.riverstyx.org/contests/ (go to River Styx 2015 Founder’s Award)
http://www.riverofwords.org/contest_rules.html (art and poetry)
Essay:
http://www.penguin.com/services-academic/essayhome/
https://www.aynrand.org/students/essay-contests
http://www.read.gov/letters/
http://thechallenge.dupont.com/essay/
Multiple categories:
http://wordsandmusic.org/william-faulkner-wisdom-competition/
http://www.theclaremontreview.ca/#!contest-submissions/c1b3x
http://blogs.colum.edu/youngauthors/submission-guidelines/
http://www.sierranevada.edu/academics/humanities-social-sciences/english/high-school-writing-contest/
https://www.writingconference.com/contest.htm
http://www.artandwriting.org/ (art and writing)
http://www.pta.org/programs/artsineducation.cfm (art and writing)
If you know of any other contests, feel free to include them in the comments!