No, Tennessee does not border the Gulf of Mexico. It is a landlocked state in the southeastern U.S., bordered by eight states: Kentucky, Virginia, North Carolina, Georgia, Alabama, Mississippi, Arkansas, and Missouri.
The Gulf Coast states are Texas, Louisiana, Mississippi, Alabama, and Florida. Tennessee's southern border touches Mississippi, Alabama, and Georgia—but stops well short of the coastline. The nearest Gulf access is via Mississippi or Alabama, roughly 150–200 miles from Tennessee's southern edge.