One suggestion is that this originated from a late Middle English contraction for 'to do off' and don is a contraction for 'to do on'.  Another suggestiion is that it comes from the Middle English word 'doffen', meaning to 'take off'.   

The term doff could also be used to mean 'to thrust aside', such as 'to doff the chamber pot'.  In other words to toss the contents out the window.  

When running on a hawt day, I've been known to doff my shirt. I make sure I'm way out in the woods, so I won't cause a spectacle.