If you buy a tri axle dump truck in Illinois and are driving it back to Massachusetts under a farm license plate do you need a CDL to drive it?

If the gross vehicle weight rating of the truck is 26,001 pounds or more, you need a CDL to operate it on the public highways. Since it's not a combination vehicle, a Class B CDL will do fine.

Additionally, farm licence plates are only valid when operated by the farmer who registered the vehicle under those plates, their immediate family, or employees of that farmer, and only within a 150 mile radius of their farm.

Not only would it be illegal for you to operate that vehicle without a CDL, but your use of a farm licence plate to transport it more than 150 miles from that farm (as well as not being a family member or employee of that farmer) is highly illegal.