What would happen if red wolves went extinct?

There is an old saying: "Nature abhors a vacuum." Any time any species goes extinct, there will be other species to fill the niche that it leaves empty. This is true for the red wolf as for any other species.
If the red wolf were to go extinct, it would upset the balance in the ecosystems in which it thrives (namely the southeastern US). This means that the animals it hunts would proliferate and over-consume the natural resources in a given area. As an area suddenly becomes devoid of resources through over-consumption, the newly proliferated animals would either die off in great numbers or migrate to other areas, which could create an imbalance there as well. 
Interestingly, red wolves were discovered to be a hybrid of gray wolves and coyotes. This is important because the gray wolf was removed from the endangered species list in certain parts of the US, such as the Northern Rocky Mountains. If future studies conclude that the red wolf is not a distinct species, it could impact future decisions as to whether the animal deserves protection.
https://www.fws.gov/southeast/pubs/alwolf.pdf

https://www.livescience.com/55586-wolves-only-one-species.html