If I win a car from the UK in a contest, and they have me go to the dealership in the U.S. to pick out colors, style interior etc., and they send the dealership the money, what type of taxes must I pay in the United States, if any? I live in Tennessee.
You would be liable for Windfall Profits Tax, I believe. http://www.irs.com/articles/windfalls-and-your-taxes
But that's paid as part of your tax return, not to anyone else.
And yeah, there are very high odds you are being scammed.