There shouldn't be any reason since the mounting holes on the MB are already insulated. Maybe they threw them in there just as an added precaution, but there should be no reason that they should be necessary.
itzdanielp is correct... there is no right or wrong answer to your question. It is save to install the motherboard with the "red" washer and it is safe to install the motherboard without the "red" washer.
IMO... the main concern is having the standoffs in the correct location to install the motherboard and that no "foreign" metal is touching the motherboard that isn't suppose to.
Some cases come with "locking" screws. They have small ridges on the bottom of the screw. If the screws you recieved with the case have the small ridges on the bottom of the head, where it contacts the motherboard,then use the red washers. If the bottom of the head is smooth, then no need to use them.
The washers protect the motherboard so that if you overtighten the screw it can destroy the area arround the mounting hole. Which could cause problems. Insufficent ground can cause lots of stability problems .
Also when you take the locking screws out a small amount of the solder arround the hole can be removed by the ridges on the bottom of the head.
Techmo is dead on . Make sure all stand offs line up with the mounting holes in the board. If the case has them already installed then remove the ones that do not line up with a mounting hole.