Classics:
* Julius Caesar by William Shakespeare
* Journey's End by R. C. Sherriff
* The Importance of Being Earnest by Oscar Wilde (though the "J" is in the title, not the beginning)
Modern:
* Jerusalem by Jez Butterworth
* Jitney by August Wilson
* The Jungle Book (various adaptations)
* Jagged Little Pill (based on the Alanis Morissette album)
* The Judas Kiss by David Hare
Musicals:
* Joseph and the Amazing Technicolor Dreamcoat by Andrew Lloyd Webber and Tim Rice
* Jesus Christ Superstar by Andrew Lloyd Webber and Tim Rice
* The Jersey Boys (about the band Frankie Valli and the Four Seasons)
For Children:
* James and the Giant Peach (adapted from the Roald Dahl book)
* The Jungle Book (various adaptations)
* The Jester's Tale (various productions)
This is just a small sampling – there are many more theatre shows beginning with the letter J out there! If you have a particular genre or theme in mind, I can offer more specific suggestions.