It was shown that Supp$(f)=\overline{D(f)}$ is true for reduced Noetherian affine schemes. But I think it is true in more general settings. Here is my proof of it for general Spec($A$).
$"\supset"$ is clearly true in general.
$"\subset"$: Suppose $p\in \overline{D(f)}$, then there is an open neighbourhood $U$ of $p$ such that $U\cap D(f)$ is empty. This means $f$ vanishes at every point of $U$. Hence $[f]_P=0$, which means $P$ isn't in Supp($f$).
I wonder if this proof is correct. Please show me a counterexample where the argument fails.
In addition, if reducedness is really essential here, I want to find a counterexample for nonreduced (but Noetherian) case.
Consider $\mathrm{Spec }(k[x,y]/(x^2,xy))$ and $f=x$.
Then $D(f)$ is empty, since every prime ideal contains $x$, but $\mathrm{Supp} (x) = \{(x,y)\} $. The problem is that $f$ vanishing at every point of $U$ does not imply that $f_P=0$.