NO, you should not include a washer. Those screws are supposed to make electrical contact with the metal on the motherboard, grounding the motherboard to the case. The multiple grounds improve stability of the system (prevents voltage gradients), and reduces the chance of a spark happening between the motherboard & case, or motherboard & PSU, should any static buildup occur.
If you do happen to use a washer, make sure it is the proper size, and is conductive (e.g. non-insulating). If the motherboard was not electrically connected to the case, there exists the chance for a voltage drop to occur between the case and motherboard, so now you have a giant capacitor (two charged plates separated between a dielectric).
TL,DR: Don't use insulated washers to prevent static discharge from happening (or use conductive ones).