I know we can't use Netflix, Hulu, or Amazon Prime on a normal basis to show movies in our classrooms because when we sign up for their service, we sign a contract stating that our accounts won't be used for public performances (personal use only). I am understanding that it is because Netflix, Hulu, and Amazon Prime do not outright own the content on their platforms and enter into agreements with the owners of the content for personal use only. However, I wanted to find out if a student can use Netflix, Hulu, or Amazon Prime to stream a portion (5 Minutes) of a movie where the student has received written permission from the Producer of a movie to show it as a public performance... I wasn't sure if the Producer permission supersedes the personal account contract of the Streaming Service?
This question is a good question for the bar exam! It is a great blend of contract law and copyright.
Well, that's enough positivity for today; time for the answer, which is...
Here’s the simply reason why I say “no”: as the member states, Netflix, Hulu, etc. tend to restrict their content for personal use, meaning: no classroom/board room viewings. This means that even if the use is "fair" or otherwise non-infringing, non-personal-use viewing is barred by the agreement the account holder has with the service.
The more sophisticated basis for me saying "no," is this: Netflix, Hulu, and their ilk bar group viewing not only because of the contractual obligations they have to their content owners, but also because to do otherwise would mess with their economic model. In short: it will cost them money. So even if a copyright owner says it's okay, they might not be inclined to consent to a use contrary to their contract.
That said, to add to the law-school-ness of your question, I'll add to your scenario:
If the student obtains a DVD or finds an online copy of the 5 minutes they need, and plays that copy (not the one from a commercial content service) to the class, if the student truly has proper permission of the copyright owner, then what would otherwise be an infringement is not.
Of course, this requires a DVD, or an online copy from a source that doesn't bar the use via contract. And of course, my scenario defeats the purpose of your question, which is to view the 5 minutes of the film in the format that is (likely) the most convenient: streaming.
I am sorry to be a bummer.
 There are of course exceptions, as these services can feature education-specific content intended for educational use. But those are the exceptions, not the general rules (at least right now).
 Which are quickly on their way to becoming as obsolete as—but not as cool as--vinyl, or cassette tapes.
 Which makes it painfully likely the copy is not 100% legit.