Sheik's shows at Brighton Music Hall in Boston; The Birchmere in Washington, DC; and World Cafe Live in Philadelphia have also been canceled.
The cancellations will allow the composer a chance to deal with an alcohol abuse problem, according to the Wall Street Journal.
"At this time, Duncan is currently struggling with an alcohol related addiction and is seeking treatment," according to a statement. "The first week of his tour has been postponed and he will be working to reschedule the dates from June 8 to June 12."
Sheik, a Grammy-winning singer-songwriter, penned the score to the Tony Award-winning musical Spring Awakening, which has book and lyrics by Steven Sater. The musical earned Tonys for Best Score, Best Book and Best Musical. Sheik's projects include The Nightingale (with Sater), as well as the new musical Whisper House (with Kyle Jarrow) and American Psycho (with Roberto Aguirre-Sacasa).