The standard examples of schemes that are not quasi-compact are either non-noetherian or have an infinite number of irreducible components. It is also easy to find non-separated irreducible examples. But are there other examples?
Question: Let $X$ be a locally noetherian scheme and assume that $X$ is irreducible (or has a finite number of irreducible components) and separated. Is $X$ quasi-compact (i.e., noetherian)?
If the answer is no in general, what conditions on $X$ are sufficient? Locally of finite type over a noetherian base scheme $S$? Fraction field finitely generated over a base? What if $X$ is regular? In general, the question is easily reduced to the case where $X$ is normal and integral.
It certainly feels like the answer is yes when $X$ is locally of finite type over $S$. Idea of proof: Choose an open dense affine $U\subseteq X$, choose a compactification $\overline{U}$ and modify $X$ and $\overline{U}$ such that the gluing $Y=X\cup_U \overline{U}$ is separated. Then, $Y=\overline{U}$ (by density and separatedness) is proper and hence quasi-compact.
Remark 1: If $X\to S$ is a proper morphism, then the irreducible components of the Hilbert scheme Hilb(X/S) are proper. The subtle point (in the non-projective case) is the quasi-compactness of the components (which can be proven by a similar trick as outlined above).
Remark 2: If $X\to S$ is universally closed, then $X\to S$ is quasi-compact. This is question 23337.
Best Answer
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).