I am not admitted in Virginia, and can only give general advice. Generally, it is not legal to cut down plants without the permission of the owner of the land. However, a governmental agency may have the right to do so for health and safety reasons. Dogwoods (cornus florida) are not considered endangered plants by the Federal government, although I am not sure about the status in Virginia. I know that in New York they are fairly common trees.
It is the state flower and tree, but it is not a protected species. You can cut it down if it is on your property.
[This communication is intended as general information and not specific legal advice, and this communication does not create an attorney-client relationship.]