Failure of your primary sponsor to fulfill this role will significantly reduce the likelihood that your project will meet its objectives.
The primary sponsor will not be demonstrating support for the project in words and actions. Those impacted by the project will not believe that it is important to the organization.
Your primary sponsor will likely underestimate or misunderstand the people side of the change.
Obstacles to addressing the people side of the change will not be addressed in a timely and effective manner.