Bastille, Chatelet les Halles, Saint Lazare, Montparnasse have tourists but these place are not touristy, the large majority of the crowd is parisian.
Barbes, Bellevillle, Faubourg Saint Denis, Porte de Choisy.
Take the subway and go in innert suburrbs : La Defense, Boulogne Billancourt, Saint Denis, Vincennes.
Districts to be avoid, you think of touristy place or dangerous places ? Per exemple many guide say to avoiding Barbes or Chatelet les Halles the sartuday but don't do that (for les Halles at least, it is the best place to see the parisian life).
For touristy places avoid Eiffel tower, Trocadero, Ile de la Cite, Saint Germain des Pres, Invalide, Rue de la Huchette.
Boulevard Saint Michel or Avenue des Champs Elysees are good even if there is many tourists, these places are always that's parisian see.