Do I remove eldritch tokens placed on the board after a mystery that placed them is solved?
According to the Reference guide, page 8:


  When the active Mystery is solved or removed from play, discard 
  all tokens on it or placed by it. Then draw a new Mystery, placing 
  it on top of any solved Mysteries.


So, in almost all cases, tokens placed my mysteries would be removed, and then replaced if the next drawn Mystery card requires additional tokens to be placed on the board.

If an individual Mystery card specifies differently, the text on the card supersedes the text in the rules.  E.g., there is a Cthulhu-specific mystery that specifies that all eldritch tokens placed as a result of that card remain even after the mystery is solved.