If you have a force.com site and let us say you want to use a do not reply email id as the from email id for your salesforce emails , salesforce does not have a clean way to implement this solution. This would be used in situations on forgot username plus password features where you dont want users to reply to an email and manage their accounts using the online self service portal. So here is the solution.
a. If all the emails are generated for a guest or anonymous user from salesforce like a forgot username/password scenario, go to force.com sites and on the detailed page click on the guest user profile. Then click on view users for the profile where system would display one guest user . Edit the user and change their email id to firstname.lastname@example.org and change their display name to a valid name. (This email id is not authenticated by salesforce for guest user at all)
b. On the customer portal settings, change the from email id to email@example.com and from name to a valid display name.
c. Any apex class which is sending an email should use the organization name on the display name and remove any other email id subjects.
a. Organization wide email address might work where you can setup an email , have it validated and then disable it.
So using this solution, you can set a do-not-reply email account for all your users on a force.com site. Please feel free to comment your thoughts on this.