I've never been required to have an abstract in any of my papers which require APA, but you should probably ask your teacher to see if it's a requirement. And if you can't contact your teacher before the paper's due, you're best off just putting out the little extra work and having one just in case. I would rather have more work done than necessary than not have the abstract and lose points if it is necessary.
I have never had to write an abstract for an essay - only for lab reports, journal papers and theses. However as Annie says, its best to check with the person who has set the assignment if you are not sure about formatting.