If I'm banned from the United States, can I fly to Mexico or Dominican Republic directly from Canada?

Anyone can be barred from entering the United States, including Canadian citizens, but that should not prevent you from overflying the country in a non-stop flight departing from Canada. A visa is not necessary, as you would not enter the United States. To do that, you have to have feet on the ground.

However, those on the No Fly List can be prevented from boarding a flight that enters US airspace, and not just by the United States; Canada has its own no-fly list, called Passenger Protect. The Canadian list incorporates data from domestic and foreign intelligence sources, including the U.S. No Fly List.

In the United States, you would make a request to the Department of Homeland Security.

DHS Traveler Redress Inquiry Program (DHS TRIP)

TSA agentsThe Department of Homeland Security Traveler Redress Inquiry Program (DHS TRIP) is a single point of contact for individuals who have inquiries or seek resolution regarding difficulties they experienced during their travel screening at transportation hubs—like airports and train stations—or crossing U.S. borders. This includes:

• watch list issues
• screening problems at ports of entry
• situations where travelers believe they have been unfairly or incorrectly delayed, denied boarding or identified for additional screening at our nation’s transportation hubs.

For Canada, it would require contacting the Passenger Protect Inquiries Office (PPIO) by either mail or email, providing it with your name and contact information, the reason for your inquiry, a detailed description of your travel issue.

By email: PS.PPinquiries-demandesPP.SP@canada.ca
By mail:
Passenger Protect Inquiries Office
PO Box 56040 Minto Place RO
Ottawa, ON K1R 7Z1

Although this question was about a different country altogether, @NateEldredge gave a great answer:

I've never heard of any situation in which you need a visa of any kind to fly over a country. You would only potentially need one if the flight was planning to land in [that country], in which case it would depend on your precise itinerary, citizenship, whether you needed to leave the transit area, etc.

Imagine the confusion if you had to apply for transit visas for every country you overflew - on a long flight there could be dozens. The international community is sensible enough not to demand that.

In theory, if your airline provided the passenger manifest to [that country] before the flight, and the ... authorities saw your name and decided they did not want you flying over their country, then they could refuse permission for the flight to enter their airspace with you on board. In that case the airline would most likely just refuse to let you board at all, so that they could fly over Russia as planned. But this is improbable; and if it did happen, having a visa wouldn't help.

If for some reason, your flight makes an unscheduled landing in Russia due to an emergency or other unexpected event, you'd either be kept on the plane, let off the plane but kept in a closed area like the airport lounge, or issued an emergency visa. You still wouldn't need to have applied for a transit visa in advance.