Have any artists that The Beatles covered also covered The Beatles?

Yes:

• The Beatles covered "Long Tall Sally", which was written and first recorded by Little Richard.  Richard covered "I Saw Her Standing There".
• The Beatles covered "I Got a Woman", which was written and first recorded by Ray Charles.  Ray covered "Yesterday", "The Long and Winding Road".  Also, he covered "Till There Was You", which was covered by The Beatles.
• The Beatles covered "You Really Got a Hold on Me", which was written and first recorded by Smokey Robinson.  Smokey, like Ray - and nearly everyone else - covered "Yesterday".