Can duct tape cut off circulation if you wrap it tightly around an injury?

Yes. Taping too tightly can cause a tourniquet effect and cut off circulation. Signs of this would be pain or numbness to the extremity or discoloration.
Acts like tourniquet. Yes, anything wrapped circumferentially around an injured extremity can cut off or greatly limit circulation and compress the nerves passing through the area. Both effects are very dangerous and risky. There is no medical indication for the use of duct tape. Seek medical attention to discuss better ways to treat your injury.