Does there exist an ideal sheaf $\mathcal F$ on some affine scheme $X$ such that $\mathcal F$ is not quasi-coherent? Please give an example such that $X$ is an affine scheme, and there exists an ideal sheaf $\mathcal F$ on $X$ such that $\mathcal F$ is not quasi-coherent.

Now that I understood the problem with my (deleted) answer, I try to write an easy example of an ideal sheaf that it is not quasicoherent: 
Let $R$ be a discrete valuation ring, with field of fractions $K$, and let $X=\operatorname {Spec}(R)=\{x,\eta\}$ be the associated affine scheme with generic point $\eta$ and closed point $x$. Consider now  the ideal sheaf $\mathcal I\subset \mathcal O_X$ defined by $\mathcal I(X)=0$ and $\mathcal I(\{\eta\})=K$. 
This ideal sheaf is not quasicoherent. 
For example, you can see that the "associated closed subscheme", in case it were quasicoherent, would correspond to the set $\{\eta\}$, which it is not closed. 
Or you can follow Samir Canning suggestion and localize at the uniformizer $\pi$ of $R$. Then $R_{\pi}=K$, $\Gamma(\operatorname{Spec}(R_{\pi}),\mathcal I)=K$, while $\Gamma(\operatorname{Spec}(R),\mathcal I)_{\pi}=0$. 
