Does the Protestant Bible mention Satan punishing non-believers after death? I have always heard people say that when you die and haven't accepted Christ and go to hell, that you're going to be punished by Satan, and we have this great plethora of contemporary culture showing a red horned devil with a trident forcing people to punishment in hell, but where does the Protestant Bible say that this is the case? 

If it doesn't say this, where did this belief originate amongst Protestant faiths?
Put simply, it doesn't.  Satan will not be the punisher in hell, but among the punished.  Satan is not any any sense the king of hell


  And when the thousand years are ended, Satan will be released from his
  prison and will come out to deceive the nations that are at the four
  corners of the earth, Gog and Magog, to gather them for battle; their
  number is like the sand of the sea. And they marched up over the broad
  plain of the earth and surrounded the camp of the saints and the
  beloved city, but fire came down from heaven[b] and consumed them, and
  the devil who had deceived them was thrown into the lake of fire and sulfur where the beast and the false prophet were, and they will
  be tormented day and night forever and ever.
  
  Revelation 20:7-10 ESV


So, the idea is a misconception.