This is a situation where you would need a court to sort it out I reckon.
Having read the statute it is very narrow:
"Nonresident" means every person who is not domiciled in the Commonwealth, except: (i) any foreign corporation that is authorized to do business in the Commonwealth by the State Corporation Commission shall be a resident of the Commonwealth for the purpose of this title; in the case of corporations incorporated in the Commonwealth but doing business outside the Commonwealth, only such principal place of business or branches located within the Commonwealth shall be dealt with as residents of the Commonwealth; (ii) a person who becomes engaged in a gainful occupation in the Commonwealth for a period exceeding 60 days shall be a resident for the purposes of this title except for the purposes of Chapter 3 (§ 46.2-300 et seq.) of this title; (iii) a person, other than a nonresident student as defined in this section, who has actually resided in the Commonwealth for a period of six months, whether employed or not, or who has registered a motor vehicle, listing an address in the Commonwealth in the application for registration shall be deemed a resident for the purposes of this title, except for the purposes of the Virginia Commercial Driver's License Act (§ 46.2-341.1 et seq.).
To cut a long story short, you've got to register it in Virginia and get a licence there going by that, so you would have to formally import the car, get the title transferred, register it, get a VA DL etc.
However these are residential ties they are forcing upon you, which can affect your immigration status as TN-1 requires non-immigrant intent.
So the question would be whether Federal law trumps State law and you would need to go to court to get an answer to that one. It does sound awfully bizarre, most States treat you as a non-resident in this situation.
Reading through the statute though insurance carriers can't issue you insurance unless the car is registered there so you have no choice, because there's no way a Canadian insurance company will insure a driver licensed in VA with a VA registered vehicle.
They need to update the statute so that the exemption for "nonresident student" is broadened to include people in a non-immigrant category.
Compare this to say, Nevada, where it is very hard indeed to show that you are resident there and need a DL.
Try writing to the AG and pointing this out.
I suspect it you went to court the ruling would be that the State can have whatever rules they want, so if you have to keep an Ontario DL to maintain residential ties to Canada, sorry, you can't drive in Virginia.
Steve.