The times shown for Start Date, Due Date and Post Date are actually in the timezone of the Turnitin server (U.S. PST). http://docs.moodle.org/24/en/Turnitin%27s_Moodle_Direct_Integration_%28Administrator_Guide%29 says:
"The integration operates in the timezone of your Moodle server, in order for Turnitin to represent the same assignment start, due and post dates you must ensure that your Turnitin account is operating in the same time zone as your Moodle server."
Instead of Turnitin's customers having to request Turnitin do a system-level mod to their account to alter the account timezone, it would be better if the plugin paid attention to the Moodle instance's time zone setting, and adjusted the dates displayed.