<p>I am building a ASP.NET website, and when I click the START DEBUGGING icon, it runs Internet Explorer as the account that I ran Visual Studio with.  </p>

<p>I would like to run IE as a different user when debugging.</p>

You can use the runas command to run IE as a different user for running your app. Later attach to the asp.net worker process when you want to debug.
If you want to just change the credentials for Integrated Authentication, try the setting in the update below.
Update
: If you want IE to prompt for a user name and password, try this setting:
For IE 7, set this option for your zone:
Tools> Internet Options > Security > custom level (for your zone)> User Authentication> Logon > prompt for user name and password