As Ash said it depends on the color of the ownership. If it is eligible for a green plate only it will say OFF ROAD on the ownership.
To answer can you get a blue plate and "just run it as a green plate" it also depends on what you mean by "run it" you can ride a blue pated bike on YOUR OWN property without insurance, (as you could with a green plate). ANYWHERE else, friends land, Ganny etc you will require insurance to operate it. Which you would also require for a green plated vehicle, if being ridden on someone elses land. If it is blue plated then your insurer will charge insurance based on that. They won't just take your word your not street ridding..lol
Pulled this from OFTR, (Ontario Federation of Trail Riders), which is the body that sells riding permits for dirt bikes.
You must have liability insurance to ride on public land including but not limited to roads, road allowances, fire roads and trails. This includes municipal forests, conservation areas and crown lands.
It is also important to note that your home policy does not cover your dirt bike in the event it is stolen or lost in a fire, flood, etc. You must have fire and theft coverage in addition to liability to cover your bike.