In general, it is permissible to reproduce portions of another person's work under the doctrine of "fair use," including applications that clearly advance education or scholarship. The Authors' Alliance provides nice introductions to the topic:
When determining "fair use," think about:
The purpose of the use
If you are making a new work using portions of copyrighted works, is the new use for an entirely different purpose? If yes, then the new purpose is transformative; e.g., in A Fair(y) Use Tale, the new purpose of the work is to inform about copyright, not to entertain -- as the original films were created to do.
If your new work using copyrighted material along with your commentary or discussion creates a whole new work designed to illuminate or create awareness of an issue, then it should be within the realm of Fair Use.
If the purpose of the use is to preserve or protect a fragile or technologically outdated format, provided that a commercial copy cannot be obtained at a "fair price." See sections 6.7-6.8 of Copyright and Cultural Institutions from Cornell University.
The nature of the copyrighted work
Especially if the nature of the copyrighted work is creative, you must make sure that your use of it results in a transformative work with a new purpose (educational, parody, etc). An example of this could be Donald Duck Meets Glenn Beck in Right Wing Radio Duck.
Just compiling a "mash-up" of Disney clips for the purpose of entertainment would result in a work of similar nature and would be an infringement of copyright.
The amount of the copying
Use only the amount necessary to achieve your new purpose for the work. Using large portions of a work - more than is absolutely necessary - to communicate your new purpose, would not be compliant.
An example of infringement would be including a whole scene of a film, when your comment or critique of it involves only one exchange of dialogue.
The effect of the copying on the market value (or potential market value) of the work
If your use of the copyrighted work may harm the market value of the work, it will not be viewed as complying with Fair Use guidelines.
For example, your use of a song to create a "mood" for your film would not be compliant. The purpose is not transformative, it is essentially used for the same purpose as the original work was created, to entertain.
Always show "good faith" by giving proper attribution to copyrighted works that you use according to the guidelines of Fair Use. See our Citation Guide for help.
A Fair(y) Use Tale Copyright explained by Disney characters in parody
"Tales from the Public Domain: Bound by Law?" Copyright in comic book form (Duke University)