Quoting from [a paper][1] by Roth and Vakil ($\mathrm{cd}$ stands for cohomological dimension):
> Let $S$ be $\mathbb{P}^2$ blown up at a point, and let $X$ be the
> affine cone over some projective embedding of $S$. Let $Z\subset X$ be
> the affine cone over the exceptional divisor of the blowup. Then $Z$
> is of codimension one in $X$, but $\mathrm{cd}(X\backslash Z) = 1$, so
> in particular it is not affine. This shows that, conversely, the
> complement of a Weil divisor in an affine scheme need not be affine...

Note that the complement of an effective Cartier divisor in an irreducible affine scheme *is* affine so if in addition to the assumptions in the question the ambient scheme is factorial ([e.g. regular][2]), then the complement of any pure codimension 1 closed subset is affine.  


  [1]: https://arxiv.org/abs/math/0406384
  [2]: https://justinsmath.wordpress.com/2014/09/23/regular-implies-locally-factorial/