I live in Erbil and I would like to know if it is possible to get a Chinese tourist visa from Erbil.

I am afraid because there is no consulate in Erbil. You may go to Chinese embassy in Baghdad to apply. The required documents are:
1. Invitation Letter of Duly Authorized Unit (Original);
2. original and copy of of your ID Card
3. original and copy of your Home Address Card (two sides; A4-size);
4. Employment Proof or Finances Proof
5. Reservation for Accommodation
6. your passport, application form and a recent six months passport photo.