I created a web application that integrates with a functionality to invite my Facebook friends. The request is sent to a friend, friend accepts the request, but on his right sidebar on the Facebook homepage, the request won't go away. The request on his Apps and Games page persists too.
Is there something I should send as callback to delete the accepted request or does Facebook do that automatically?
Yes you have to delete the request, see the documentation here:
https://developers.facebook.com/docs/requests/#deleting https://developers.facebook.com/docs/howtos/requests#delete_requests