Determining when permission is required to perform a play, who to ask for permission and what kind of rights are needed begins with a complete understanding of the play and the performance, including who your performers are (amateur vs. professional), the venue and the audience, the number of performances, and whether or not music is involved.
To begin, if you and/or your students are performing any kind of work, including a play or a musical, as part of a course at LSU and only other students enrolled in the course are performing as well as viewing the performance, the copyright law (17 U.S.C. Section 110(1)) permits this without requiring any sort of license or performance fee. Additionally, works that have entered the public domain, e.g., Shakespeare, are free to use.
Beyond that, you should assume that you need permission, most likely through a license, for a proposed public performance of a play, musical, or other theater performance. The specifics of your production will dictate whether and how much this license will cost. Keep in mind that licenses to publicly perform a play usually do not include copies of the script (or anything else) nor do they include a grand performance right as discussed under the Music Copyright section.
Here are some helpful resources:
• Obtaining rights to produce a play or musical (Yale, part way down page adapationg of Rachel Durkin's original piece)
• Obtaining Copyright Permissions: Theatre (libguides UMichigan Library)
• Statement of Best Practices In Fair Use of Dance-Related Materials
Sources of Licenses:
• Samuel French [appears to contain the Baker’s Plays] musical and non musical theater, especially the FAQ
• Dramatists Play Service (DPS), especially their FAQ