example of a scheme where connected components aren't open In general topology connected components are open if there are finitely many of them, but otherwise may not be. For example, the connected components of $\mathbb{Q}\subset\mathbb{R}$ are the singletons, which are closed but not open.
What are some interesting examples of schemes where connected components aren't open?
 A: I don't know about interesting, but there are examples. Let $A = \mathbb{F}_2^{\mathbb{N}}$, i.e. the cartesian product of countably many copies of $\mathbb{F}_2$ and let $X = \operatorname{Spec} A$. I claim $X$ is the ultrafilter space $\beta \mathbb{N}$, also known as the Stone–Čech compactification of $\mathbb{N}$.
By construction, $A$ is a boolean ring, corresponding to the boolean algebra $\mathscr{P} (\mathbb{N})$, so maximal ideals of $A$ are the same as ultrafilters on $\mathbb{N}$. Moreover, since quotients of boolean rings are boolean rings, if $\mathfrak{p}$ is a prime ideal of $A$, then $A / \mathfrak{p}$ is a boolean integral domain – but the only such thing is $\mathbb{F}_2$, so $\mathfrak{p}$ is also a maximal ideal. 
Thus, there is a canonical bijection between the points of $X$ and the set of ultrafilters on $\mathbb{N}$. It is not hard to see that the Zariski topology on $X$ corresponds to the Stone topology on $\beta \mathbb{N}$, so we have a homeomorphism. But the latter is known to be totally disconnected and not discrete, so $X$ is a scheme whose connected components are not open.
