Are you sure it's obvious when $X$ is affine? Try working out the example where $X=\mathbb{A}^1$, $Y=\{0\}$, and $F$ is $\mathcal{O}_Y$ (considered as a sheaf on $X$).
For a more obvious (to me) but less geometric counterexample, consider what happens to the exact sequence of $\mathbb{Z}$-modules $ 0 \to 2\mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0 $ when you take tensor products with $\mathbb{Z}/2\mathbb{Z}$.
The property you're interested in is flatness: an $R$-module $M$ is flat precisely when the operation of taking tensor products with $M$ preserves exactness. This turns out to be equivalent to your sequence being exact for all ideals $I$.