Praise your students for effort rather than perfection. Help them divorce their self-worth from their scores and focus it on intangible things like improvement, hard work and motivation. Encourage your students to tackle intimidating tasks, knowing that they might not succeed, and then praise them for their courage and ambition.
Assign projects that require imagination and creativity rather than precision, and don't grade them. Give students the opportunity to work on something which they cannot do "wrong." Evaluate their work using language that rewards effort rather than perfection, such as "interesting ideas" instead of "perfect job.".
Talk about your own errors comfortably and without shame. Show your students that everyone makes mistakes every day, including particularly smart and admirable people. Point out that even famously successful individuals do not succeed at everything they do -- Babe Ruth struck out over a thousand times, Thomas Edison made a thousand light bulb prototypes that didn't work, Walt Disney was fired from a newspaper job for having no ideas, and so on.
Help your students change the belief that that they are stupid, worthless or a failure if they make a mistake. For example, if a student believes she cannot spell because she missed one word on a spelling test, point out that she learned the other words in time for the test -- and now she will probably remember the correct spelling for the missed word as well. Thus, her "failure" actually led to her learning the word successfully.
Teach your students time management for long projects and multiple assignments. Help them prioritize bigger assignments over smaller ones, for example, by reminding them that having good ideas for their essay is more important than having correct punctuation in a 10-minute journal entry. Show them how to break big assignments into smaller pieces so that they don't seem so overwhelming.
Give timed assignments to students whose perfectionism stops them from completing projects. Tell them that the purpose of the assignment is for them to finish it on time, not to get every detail exactly right. Explain that "succeeding" on these projects means not procrastinating or agonizing rather than not making any mistakes.
Discuss the advantages and disadvantages of perfectionism with your students. Point out that equating success with perfection makes them unhappy and stops them from appreciating their achievements. Encourage students to record their feelings and experiences in a journal that will not be graded or evaluated by anyone else.
Suggest that your students find an extra-curricular activity with no scores, grades or stakes attached. Help them find something they simply enjoy doing. A hobby without consequences can help them relax, give them an outlet for frustration or cheer them up when they become depressed.