There are smooth counterexamples. Let $S_0$ be a smooth separated irreducible scheme over a field $k$ with dimension $n > 0$, and $s_0 \in S_0(k)$. Blow up $s_0$ to get another such scheme $S_1$ with a $\mathbf{P}^{n-1}_k$ over $s_0$. Blow up a $k$-point $s_1$ over $s_0$ to get $S_2$, and keep going. Get pairs $(S_n, s_n)$ so that the open complement $U_n$ of $s_n$ in $S_n$ is open in $U_ {n+1}$ and is strictly contained in it. Glue them together in the evident manner, to get a smooth irreducible $k$-scheme. It is locally of finite type, but is not quasi-compact (since the $U_n$ are an open cover with no finite subcover). This is separated (either by direct consideration of affine open overlaps, or by using the valuative criterion).

There are smooth counterexamples. Let $S_0$ be a smooth separated irreducible scheme over a field $k$ with dimension $d > 1$, and $s_0 \in S_0(k)$. Blow up $s_0$ to get another such scheme $S_1$ with a $\mathbf{P}^{d-1}_k$ over $s_0$. Blow up a $k$-point $s_1$ over $s_0$ to get $S_2$, and keep going. Get pairs $(S_n, s_n)$ so that the open complement $U_n$ of $s_n$ in $S_n$ is open in $U_ {n+1}$ and is strictly contained in it. Glue them together in the evident manner, to get a smooth irreducible $k$-scheme. It is locally of finite type, but is not quasi-compact (since the $U_n$ are an open cover with no finite subcover). This is separated (either by direct consideration of affine open overlaps, or by using the valuative criterion).