None, unless the person actually violates existing laws and then he or she will experience the same penalties as anyone else.
Renouncing citizenship will not lead to deportation or the FBI knocking at the door or any other such action. Quite frankly it's doubtful anyone would care except perhaps those persons close to the renouncer and maybe their employer although it's unlikely an anarchist would choose to work in a corrupt society.
Of course if the anarchist begins making threats upon federal, state or local government or govermental officials his or her status will change rather rapidly.
Declare away! Make yourself a king, despot, saint, sinner or leader of the down trodden, whatever does it for you.